hypercompositional algebra and applications

Edited by

Hypercompositional Algebra and Applications

Christos G MassourosPrinted Edition of the Special Issue Published in Mathematics

wwwmdpicomjournalmathematics

Hypercompositional Algebra andApplications


Editor

Christos G Massouros

MDPI bull Basel bull Beijing bullWuhan bull Barcelona bull Belgrade bullManchester bull Tokyo bull Cluj bull Tianjin

Editor


Core Department

National and Kapodistrian

University of Athens

Athens

Greece

Editorial Office

MDPI

St Alban-Anlage 66

4052 Basel Switzerland

This is a reprint of articles from the Special Issue published online in the open access journal

Mathematics (ISSN 2227-7390) (available at wwwmdpicomjournalmathematicsspecial issues

Hypercompositional Algebra Applications)

For citation purposes cite each article independently as indicated on the article page online and as

indicated below

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Page Range

ISBN 978-3-0365-2104-6 (Hbk)

ISBN 978-3-0365-2103-9 (PDF)

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license CC BY-NC-ND

Contents

About the Editor vii

Preface to rdquoHypercompositional Algebra and Applicationsrdquo ix

Christos Massouros and Gerasimos MassourosAn Overview of the Foundations of the Hypergroup TheoryReprinted from Mathematics 2021 9 1014 doi103390math9091014 1

Gerasimos Massouros and Christos MassourosHypercompositional Algebra Computer Science and GeometryReprinted from Mathematics 2020 8 1338 doi103390math8081338 43

Milica Kankaras and Irina CristeaFuzzy Reduced HypergroupsReprinted from Mathematics 2020 8 263 doi103390math8020263 75

Hashem Bordbar and Irina CristeaRegular Parameter Elements and Regular Local HyperringsReprinted from Mathematics 2021 9 243 doi103390math9030243 87

Sarka Hoskova-Mayerova Madeline Al Tahan and Bijan DavvazFuzzy Multi-HypergroupsReprinted from Mathematics 2020 8 244 doi103390math8020244 101

Jan Chvalina Michal Novak Bedrich Smetana and David StanekSequences of Groups Hypergroups and Automata of Linear Ordinary Differential OperatorsReprinted from Mathematics 2021 9 319 doi103390math9040319 115

Dariush Heidari and Irina CristeaOn Factorizable SemihypergroupsReprinted from Mathematics 2020 8 1064 doi103390math8071064 131

Mario De Salvo Dario Fasino Domenico Freni and Giovanni Lo Faro1-Hypergroups of Small SizesReprinted from Mathematics 2021 9 108 doi103390math9020108 139

Stepan Krehlıkn-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control ofLane ShiftingReprinted from Mathematics 2020 8 835 doi103390math8050835 157

Vahid Vahedi Morteza Jafarpour Sarka Hoskova-Mayerova Hossein Aghabozorgi VioletaLeoreanu-Fotea and Svajone BekesieneDerived Hyperstructures from HyperconicsReprinted from Mathematics 2020 8 429 doi103390math8030429 175

Christos G Massouros and Naveed YaqoobOn the Theory of LeftRight Almost Groups and Hypergroups with their RelevantEnumerationsReprinted from Mathematics 2021 9 1828 doi103390math9151828 191

v

About the Editor


Christos G Massouros is a Full Professor of Mathematics at the National and Kapodistrian

University of Athens Christos G Massouros is a graduate of the National and Kapodistrian

University of Athens Department of Mathematics He did graduate work in theoretical physics and

mathematics at the National Center for Scientific Research ldquoDemocritusrdquo of Greece and he received

his PhD from the School of Applied Mathematical and Physical Sciences of the National Technical

University of Athens His main research interests are in the theory of hypercompositional structures

and their applications He is the author of a number of research review and survey papers on

the subject and has participated in many relevant research programs He is a reviewer for several

scientific journals and member of their editorial boards

vii

Preface to rdquoHypercompositional Algebra andApplicationsrdquo

This Special Issue is about Hypercompositional Algebra which is a recent branch of Abstract

Algebra

As it is known Algebra is a generalization of Arithmetic where Arithmetic (from the Greek word

mdasharithmosmdashmeaning ldquonumberrdquo) is the area of mathematics that deals with numbers During the

classical period Greek mathematicians created a Geometric Algebra where terms were represented

by sides of geometric objects while the Alexandrian School that followed (which was founded

during the Hellenistic era) changed this approach dramatically Especially Diophantus made the

first fundamental step towards Symbolic Algebra as he developed a mathematical notation in order

to write and solve algebraic equations

In the Middle Ages the stage of mathematics shifted from the Greek world to the Arabic

and in AD 830 the Persian mathematician Muammad ibn Musa al-Khwarizmı wrote his famous

treatise al-Kitab al-mukhtaar fı isab al-jabr warsquol-muqabala which contains the first systematic solution

of linear and quadratic equations The word ldquoAlgebrardquo derives from this bookrsquos title (al-jabr)

After the 15th century AD the stage of mathematics changed again and went back to Europe

leaving the Islamic world which was in decline while the European world was ascending as the

Renaissance began to spread in Western Europe However Algebra essentially remained Arithmetic

with non-numerical mathematical objects until the 19th century when the algebraic mathematical

thought changed radically due to the work of two young mathematicians the Norwegian Niels

Henrik Abel (1802ndash1829) and the Frenchman Evariste Galois (1811ndash1832) on the algebraic equations

It was then understood that the same processes could be applied to various objects or sets of entities

other than numbers Therefore Abstract Algebra was born

The early years of the next (20th) century brought the end of determinism and certainty to

science The uncertainty affected Algebra as well and was expressed in the work of the young French

mathematician Frederic Marty (1911ndash1940) who introduced an algebraic structure in which the rule

of synthesizing elements gives a set of elements instead of one element only He called this structure

ldquohypergrouprdquo and he presented it during the 8th congress of Scandinavian Mathematicians held in

Stockholm in 1934

Unfortunately Marty was killed at the age of 29 when his airplane was hit over the

Baltic Sea while he was in military duty during World War II His mathematical heritage on

hypergroups is three papers only However other mathematicians such as H Wall M Dresher

O Ore M Krasner and M Kuntzmann started working on hypergroups shortly thereafter Thus

the Hypercompositional Algebra came into being as a branch of Abstract Algebra that deals

with structures endowed with multi-valued operations Such operations which are also called

ldquohyperoperationsrdquo or ldquohypercompositionsrdquo are laws of synthesis of the elements of a nonempty set

which associate a set of elements instead of a single element with every pair of elements

Nowadays this theory is characterized by huge diversity of character and content and can

present results in mathematics and other sciences such as computer science information technologies

physics chemistry biology social sciences etc

This Special Issue titled ldquoHypercompositional Algebra and Applicationsrdquo contains 11

peer-reviewed papers written by 24 outstanding experts in the theory of hypercompositional

structures The names of the authors are in alphabetical order

ix

Hossein Aghabozorgi Madeline Al Tahan Svajone Bekesiene Hashem Bordbar Jan Chvalina

Irina Cristea Bijan Davvaz Mario De Salvo Dario Fasino Domenico Freni Dariush Heidari Sarka

Hoskova-Mayerova Morteza Jafarpour Milica Kankaras Stepan Krehlık Violeta Leoreanu-Fotea

Giovanni Lo Faro Christos G Massouros Gerasimos Massouros Michal Novak Bedrich Smetana

David Stanek Vahid Vahedi and Naveed Yaqoob

The introductory paper in this Issue is a review paper on the foundations of the hypergroup

theory that focuses on the presentation of the essential principles of the hypergroup which is

the prominent structure of the Hypercompositional Algebra The 20th century was not only the

period when the hypergroup came into being but it was also characterized by the multi-volume

and extremely valuable work Elements de mathematique by Nicolas Bourbaki where the notion

of the mathematical structure an idea related to the broader and interdisciplinary concept of

structuralism was introduced Towards this direction the paper generalizes the notion of the

magma which was introduced in Algebre by Nicolas Bourbaki (1943 by Hermann Paris) in order

to include the hypercompositional structures and thereafter it reveals the structural relation between

the fundamental entity of the Abstract Algebra the group with the hypergroup as it shows that both

structures satisfy exactly the same axioms but under different synthesizing laws

I hope that this Special Issue will be useful not only to the mathematicians who work on the

Hypercompositinal Algebra but to all scientists from different disciplines who study the Abstract

Algebra and seek its applications


Editor

x

mathematics

Review

An Overview of the Foundations of the Hypergroup Theory

Christos Massouros 1 and Gerasimos Massouros 2

Citation Massouros C Massouros

G An Overview of the Foundations

of the Hypergroup Theory

Mathematics 2021 9 1014 https

doiorg103390math9091014

Academic Editor Michael Voskoglou

Received 21 February 2021

Accepted 16 March 2021

Published 30 April 2021

Publisherrsquos Note MDPI stays neutral

with regard to jurisdictional claims in

published maps and institutional affil-

iations

Copyright copy 2021 by the authors

Licensee MDPI Basel Switzerland

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https

creativecommonsorglicensesby

40)

1 Core Department Euripus Campus National and Kapodistrian University of AthensGR 34400 Euboia Greece

2 School of Social Sciences Hellenic Open University Aristotelous 18 GR 26335 Patra Greecegermasourosgmailcom

Correspondence ChrMasuoagr or ChMassourosgmailcom

Abstract This paper is written in the framework of the Special Issue of Mathematics entitled ldquoHyper-compositional Algebra and Applicationsrdquo and focuses on the presentation of the essential principlesof the hypergroup which is the prominent structure of hypercompositional algebra In the beginningit reveals the structural relation between two fundamental entities of abstract algebra the groupand the hypergroup Next it presents the several types of hypergroups which derive from theenrichment of the hypergroup with additional axioms besides the ones it was initially equippedwith along with their fundamental properties Furthermore it analyzes and studies the varioussubhypergroups that can be defined in hypergroups in combination with their ability to decomposethe hypergroups into cosets The exploration of this far-reaching concept highlights the particularityof the hypergroup theory versus the abstract group theory and demonstrates the different techniquesand special tools that must be developed in order to achieve results on hypercompositional algebra

Keywords group hypergroup subhypergroup cosets

1 Introduction

The early years of the 20th century brought the end of determinism and certainty toscience The emergence of quantum mechanics rocked the well-being of classical mechanicswhich was founded by Isaac Newton in Philosophiaelig Naturalis Principia Mathematica In 1927Werner Heisenberg developed his uncertainty principle while working on the mathematicalfoundations of quantum mechanics On the other hand in 1931 Kurt Goumldel published histwo incompleteness theorems thus giving an end to David Hilbertrsquos mathematical dreamsand to the attempts that are culminating with Principia Mathematica of Bertrand RussellIn 1933 Andrey Kolmogorov published his book Foundations of the Theory of Probabilityestablishing the modern axiomatic foundations of probability theory In the same decadethe uncertainty invaded algebra as well A young French mathematician Freacutedeacuteric Marty(1911ndash1940) during the 8th Congress of Scandinavian Mathematicians held in Stockholmin 1934 introduced an algebraic structure in which the rule of synthesizing elements resultsto a set of elements instead of a single element He called this structure hypergroupMarty was killed at the age of 29 when his airplane was hit over the Baltic Sea whilehe was in the military during World War II His mathematical heritage on hypergroupswas only three papers [1ndash3] However other mathematicians such as M Krasner [4ndash8]J Kuntzmann [8ndash10] H Wall [11] O Ore [12ndash14] M Dresher [13] E J Eaton [1415] andL W Griffiths [16] gradually started working on hypergroups shortly thereafter (see theclassical book [17] for further bibliography) Thus hypercompositional algebra cameinto being

Hypercompositional algebra is the branch of abstract algebra that deals with structuresequipped with multivalued operations Multivalued operations also called hyperoper-ations or hypercompositions are laws of synthesis of the elements of a nonempty setwhich associates a set of elements instead of a single element to every pair of elements

1

Mathematics 2021 9 1014

The fundamental structure of hypercompositional algebra is the hypergroup This paperenlightens the structural relation between the groups and the hypergroups The study ofsuch relationships is at the heart of structuralism Structuralism is based on the idea thatthe elements of a system under study are not important and only the relationships andstructures among them are significant As it is proved in this paper the axioms of groupsand hypergroups are the same while these algebraic entitiesrsquo difference is based on therelationship between their elements which is created by the law of synthesis In groupsthe law of synthesis of any two elements is a composition ie a single element while inhypergroups it is a hypercomposition that is a set of elements

The next section of this paper generalizes the notion of magma which was introducedin Eacuteleacutements de Matheacutematique Algegravebre [18] by Nicolas Bourbaki and so will include algebraicstructures with hypercomposition The third section presents a unified definition of thegroup and the hypergroup This definition of the group is not included in any grouptheory book and its equivalence to the already-known ones is proved in the fourth sectionThe fifth section presents another equivalent definition of the hypergroup while certainof its fundamental properties are proved As these properties derive directly from theaxioms of the hypergroup they outline the strength of these axioms So for instance it isshown that the dominant in the bibliography definition of the hypercomposition includesredundant assumptions The restriction that a hypercomposition is a mapping from Etimes Einto the family of nonempty subsets of E is needless since in the hypergroups the resultof the hypercomposition is proved to be always a nonvoid set The sixth section dealswith different types of hypergroups The law of synthesis imposes a generality on thehypergroup which allows its enrichment with more axioms This creates a multitude ofspecial hypergroups with many and interesting properties and applications The join spaceis one of them It was introduced by W Prenowitz in order to study geometry with the toolsof hypercompositional algebra and many other researchers adopted this approach [19ndash34]Another one is the fortified join hypergroup which was introduced by G Massouros inhis study of the theory of formal languages and automata [35ndash42] and he was followedby other authors who continued in this direction eg [42ndash52] One more is the canonicalhypergroup which is the additive part of the hyperfield that was used by M Krasner asthe proper algebraic tool in order to define a certain approximation of complete valuedfields by sequences of such fields [53] This hypergroup was used in the study of geometryas well eg [323354ndash60] Moreover the canonical hypergroup became part of otherhypercompositional structures like the hypermodule [61] and the vector hyperspace [62]In [61] it is shown that analytic projective geometries and Euclidean spherical geometriescan be considered as special hypermodules Furthermore the hyperfields were connectedto the conic sections via a number of papers [63ndash65] where the definition of an ellipticcurve over a field F was naturally extended to the definition of an elliptic hypercurveover a quotient Krasnerrsquos hyperfield The conclusions obtained in [63ndash65] can be appliedto cryptography as well Moreover D Freni in [66] extended the use of the hypergroupin more general geometric structures called geometric spaces [67] contains a detailedpresentation of the above Also hypergroups are used in many other research areas likethe ones mentioned in [68] and recently in social sciences [69ndash73] and in an algebraizationof logical systems [7475] The seventh section refers to subhypergroups A far-reachingconcept of abstract group theory is the idea of the decomposition of a group into cosetsby any of its subgroups This concept becomes much more complicated in the case ofhypergroups The decomposition of the hypergroups cannot be dealt with in a similaruniform way as in the groups So in this section and depending on its specific typethe decomposition of a hypergroup to cosets is treated with the use of invertible closedreflexive or symmetric subhypergroups

Special notation In the following in addition to the typical algebraic notations we useKrasnerrsquos notation for the complement and difference So we denote with A middot middotB the set ofelements that are in the set A but not in the set B

2


2 Magma

In Eacuteleacutements de Matheacutematique Algegravebre [18] Nicolas Bourbaki used the Greek wordmagma which comes from the verb microασσω (= ldquokneadrdquo) to indicate a set with a law ofcomposition The following definition extends this notion in order to incorporate moregeneral laws of synthesizing the elements in a set

Definition 1 Let E be a nonvoid set A mapping from Etimes E into E is called a composition on Eand a mapping from Etimes E into the power set P(E) of E is called a hypercomposition on E A setwith a composition or a hypercomposition is called a magma

The notation (Eperp) where perp is the composition or the hypercomposition is usedwhen it is required to write the law of synthesis in a magma The image perp(x y) of (x y)is written xperpy The symbols + and middot are the most commonly used instead of perp A lawof synthesis denoted by the symbol + is called addition and x + y is called the sum of xand y if the synthesis is a composition and the hypersum of x and y if the synthesis is ahypercomposition A law of synthesis denoted by the symbol middot is called multiplication andx middot y is called the product of x and y if the synthesis is a composition and the hyperproduct ofx and y if the synthesis is a hypercomposition when there is no likelihood of confusionthe symbol middot can be omitted and we write xy instead of x middot y

Example 1 The power set P(E) of a set E 6= empty is a magma if

(X Y)rarr X cupY or (X Y)rarr X capY

Example 2 The set N of natural numbers is a magma under addition or multiplication

Example 3 A nonvoid set E becomes a magma under the following law of synthesis

(x y)rarr x y

The above law of synthesis is called b-hypercomposition E also becomes a magma if

(x y)rarr E

This law of synthesis is called total hypercomposition

Two significant types of hypercompositions are the closed and the open ones A hyper-composition is called closed [76] (or containing [77] or extensive [78]) if the two participatingelements always belong to the result of the hypercomposition while it is called open ifthe result of the hypercomposition of any two different elements does not contain thesetwo elements

Example 4 Let S be the set of points of a Euclidian geometry In S we define the following law ofsynthesis ldquomiddotrdquo if a and b are distinct points of S then a middot b is the set of all elements of the segmentab while a middot a is taken to be the point a for any point a of S (Figure 1) Then the set S of the pointsof the Euclidian geometry becomes a magma Usually a middot b is written simply as ab and it is calledthe join of a and b It is worth noting that we can actually define two laws of synthesis an open anda closed hypercomposition depending on whether we consider the open or the closed segment ab

Mathematics 2021 9 x FOR PEER REVIEW 3 of 45

2 Magma

In Eacuteleacutements de Matheacutematique Algegravebre [18] Nicolas Bourbaki used the Greek word

magma which comes from the verb μάσσω (= ldquokneadrdquo) to indicate a set with a law of

composition The following definition extends this notion in order to incorporate more

general laws of synthesizing the elements in a set

Definition 1 Let E be a nonvoid set A mapping from E E into E is called a composition on E and a mapping from E E into the power set P E of E is called a hypercomposition

on E A set with a composition or a hypercomposition is called a magma

The notation E where is the composition or the hypercomposition is used

when it is required to write the law of synthesis in a magma The image x y of x y

is written x y The symbols + and are the most commonly used instead of A law

of synthesis denoted by the symbol + is called addition and x y is called the sum of x and y if the synthesis is a composition and the hypersum of x and y if the synthesis is a hypercomposition A law of synthesis denoted by the symbol is called multiplica‐

tion and x y is called the product of x and y if the synthesis is a composition and the

hyperproduct of x and y if the synthesis is a hypercomposition when there is no like‐

lihood of confusion the symbol can be omitted and we write xy instead of x y

Example 1 The power set P E of a set E is a magma if

X Y X Y or X Y X Y

Example 2 The set of natural numbers is a magma under addition or multiplication


x y x y

The above law of synthesis is called b‐hypercomposition E also becomes a magma if

x y E


Two significant types of hypercompositions are the closed and the open ones A hy‐

percomposition is called closed [76] (or containing [77] or extensive [78]) if the two partici‐

pating elements always belong to the result of the hypercomposition while it is called

open if the result of the hypercomposition of any two different elements does not contain

these two elements

Example 4 Let S be the set of points of a Euclidian geometry In S we define the following law of syn‐

thesis ldquo rdquo if a and b are distinct points of S then a b is the set of all elements of the segment ab while a a is taken to be the point a for any point a of S (Figure 1) Then the set S of the

points of the Euclidian geometry becomes a magma Usually a b is written simply as ab and it is called the join of a and b It is worth noting that we can actually define two laws of synthesis an

open and a closed hypercomposition depending on whether we consider the open or the closed

segment ab

Figure 1 Example 4

Any finite magma E can be explicitly defined by its synthesis table If E consists of nelements then the synthesis table is a ntimes n square array heading both to the left and above

3


by a list of the n elements of E In this table (Cayley table) the entry in the row headed byx and the column headed by y is the synthesis xperpy

Example 5 Suppose that E = 1 2 3 4 Then the law of the synthesis is a composition in thefirst table and a hypercomposition in the second

perp 1 2 3 4

1 1 3 2 32 3 2 1 43 1 2 1 14 2 3 4 1

perp 1 2 3 4

1 123 13 24 1342 3 1234 13 243 123 12 1234 14 23 23 34 13

Let (Eperp) be a magma Given any two nonvoid subsets X Y of E then

XperpY = xperpy isin E | x isin X y isin Y if perp is a composition

and XperpY = cupxisinX yisinY

(xperpy) if perp is a hypercomposition

If X or Y is empty then XperpY is empty If a isin E we usually write aperpY instead of aperpYand Xperpa instead of Xperpa In general the singleton a is identified with its member aSometimes it is convenient to use the relational notation A asymp B to assert that subsets Aand B have a nonvoid intersection Then as the singleton a is identified with its membera the notation a asymp A or A asymp a is used as a substitute for a isin A The relation asymp may beconsidered as a weak generalization of equality since if A and B are singletons and A asymp Bthen A = B Thus a asymp bperpc means a = bperpc if the synthesis is a composition and a isin bperpcif the synthesis is a hypercomposition

Definition 2 Let (Eperp) be a magma The law of synthesis

(x y)rarr xperpopy = yperpx

is called the opposite of perp The magma (Eperpop) is called the opposite magma of (Eperp) Whenperpop = perp the law of synthesis is called commutative and the magma is called commutative magma

Definition 3 Let E be a magma and X a subset of E The set of elements of E that commute witheach one of the elements of X is called the centralizer of X The centralizer of E is called the centerof E An element of the center of E is called the central element of E

Every law of synthesis in a magma induces two new laws of synthesis If the law ofsynthesis is written multiplicatively then the two induced laws are

ab = x isin E | a asymp xb and ba = x isin E | a asymp bx

Thus x asymp ab if and only if a asymp xb and x asymp ba if and only if a asymp bx In the case of amultiplicative magma the two induced laws are named inverse laws and they are called theright and left division respectively It is obvious that if the magma is commutative then theright and left divisions coincide

4


Proposition 1 If the law of synthesis in a magma (E middot) is an open hypercomposition thenaa = aa = a for all a in E while aa = aa = E for all a in E when the law of synthesis is aclosed (containing) hypercomposition

Example 6 On the set Q of rational numbers multiplication is a commutative law of synthesisThe inverse law is as follows

ab =

ab i f b 6= 0

empty i f b = 0 a 6= 0

Q i f b = 0 a = 0

Here the law of synthesis is a composition and the inverse law is a hypercomposition

Example 7 In Example 4 a law of synthesis was defined on the set S of the points of a Euclidiangeometry If we consider the open hypercomposition then the inverse law is the following If a andb are two distinct points in S then ab is the set of the points of the open halfline with endpointa that is opposite to point b while aa = a for any point a of S (Figure 2) Usually this law iscalled the extension of a over b or ldquoa from brdquo


called the right and left division respectively It is obvious that if the magma is commuta‐

tive then the right and left divisions coincide

Proposition 1 If the law of synthesis in a magma E is an open hypercomposition then

a a a a a for all a in E while a a a a E for all a in E when the law of synthesis is a closed (containing) hypercomposition

Example 6 On the set ℚ of rational numbers multiplication is a commutative law of synthesis The inverse law is as follows

0

0 0

0 0

aif b

b

a bif b a

if b a


Example 7 In Example 4 a law of synthesis was defined on the set S of the points of a Euclidian

geometry If we consider the open hypercomposition then the inverse law is the following If a and b are two distinct points in S then a b is the set of the points of the open halfline with

endpoint a that is opposite to point b while a a a for any point a of S (Figure 2) Usually this law is called the extension of a over b or ldquo a from b rdquo

Figure 2 Example 7

Example 8 Let E

be a magma and let and be the right and left division A new law of

synthesis called the extensive enlargement of can be defined on E as follows

a b a b a b for all a b E

Denoting the two induced laws of synthesis by and it is immediate that

a b a if a ba b

E if a b

and

b a a if a bb a

E if a b

Obviously the extensive enlargement is a closed (containing) hypercomposition

Definition 4 An element 0 of a magma E is called left absorbing (resp right absorbing)

if 0 0x (resp 0 0x ) for all x E An element of a magma is called absorbing if it is a

bilaterally absorbing element

A direct consequence of the above definition is Proposition 2

Proposition 2 If a magma has a left (resp right) absorbing element then the relevant induced

law of synthesis is a hypercomposition

Definition 5 A law of synthesis x y x y on a set E is called associative if the property

Figure 2 Example 7

Example 8 Let (E middot) be a magma and let and be the right and left division A new law ofsynthesis called the extensive enlargement of ldquomiddotrdquo can be defined on E as follows

a b = a middot b cup a b f or all a b isin E


a b =

ab cup a i f a 6= bE i f a = b

and ba =

ba cup a i f a 6= bE i f a = b


Definition 4 An element 0 of a magma (Eperp) is called left absorbing (resp right absorbing) if0perpx = 0 (resp xperp0 = 0) for all x isin E An element of a magma is called absorbing if it is abilaterally absorbing element


Proposition 2 If a magma has a left (resp right) absorbing element then the relevant induced lawof synthesis is a hypercomposition

Definition 5 A law of synthesis (x y)rarr xperpy on a set E is called associative if the property

(xperpy)perpz = xperp(yperpz)

is valid for all elements x y z in E A magma whose law of synthesis is associative is called anassociative magma

5


Example 9 If the law of synthesis is the b-hypercomposition denoted by + then

(x + y) + z = x y+ z = (x + z) cup (y + z) = x z cup y z = x y z

and x + (y + z) = x + y z = (x + y) cup (x + z) = x y cup x z = x y z

Hence (x + y) + z = x + (y + z) for all x y z in E Thus the b-hypercomposition is associative

Example 10 The law of synthesis defined in Example 4 on the set S of the points of a Euclidiangeometry is associative For the verification it is required to consider many cases The followingFigure 3 presents the general case for the open hypercomposition in which the points a b c are notcollinear The result of both a(bc) and (ab)c is the interior of the triangle abc


x y z x y z

is valid for all elements x y z in E A magma whose law of synthesis is associative is called an

associative magma

Example 9 If the law of synthesis is the b‐hypercomposition denoted by + then

x y z x y z x z y z x z y z x y z

and x y z x y z x y x z x y x z x y z

Hence x y z x y z for all x y z in E Thus the b‐hypercomposition is associative

Example 10 The law of synthesis defined in Example 4 on the set S of the points of a Euclidian

geometry is associative For the verification it is required to consider many cases The following

Figure 3 presents the general case for the open hypercomposition in which the points a b c are

not collinear The result of both a bc and ab c is the interior of the triangle abc

Figure 3 Example 10

The interaction of the law of synthesis with the two induced laws in an associative

magma gives the mixed associativity

Proposition 3 In an associative magma E the properties

a b c a cb right mixed associativity

c b a bc a left mixed associativity

c bb aa c

are valid for all a b c E

Proof Let x a b c Then we have the following sequence of equivalent statements

x a b c xc a b a xc b a x cb x a cb

therefore a b c a cb Similar is the proof of the left mixed associativity Next let

x b a c Then we have the sequence of implications

x c xc b a a b xc a bx c bx a c x bb aa c

Figure 3 Example 10

The interaction of the law of synthesis with the two induced laws in an associativemagma gives the mixed associativity

Proposition 3 In an associative magma (E middot) the properties

(ab)c = a(cb) right mixed associativityc(ba) = (bc)a le f t mixed associativity(ba)c = b(ac)

are valid for all a b c isin E

Proof Let x asymp (ab)c Then we have the following sequence of equivalent statements

x asymp (ab)chArr xc asymp abhArr a asymp (xc)bhArr a asymp x(cb)hArr x asymp acb

therefore (ab)c = a(cb) Similar is the proof of the left mixed associativity Next letx asymp (ba)c Then we have the sequence of implications

x asymp (ba)chArr xc asymp bahArr a asymp b(xc)hArr a asymp (bx)chArr bx asymp achArr x asymp b(ac)

hence (ba)c = b(ac)

Corollary 1 In an associative magma (E middot) the equalities

(AB)C = A(CB)C(BA) = (BC)A(BA)C = B(AC)

are valid for all nonvoid subsets A B C of E

6


Proposition 4 In an associative magma (E middot) it holds that

b asymp (ab)a and b asymp a(ba)

for all a b isin E

Proof Let x asymp ab Then a asymp xb therefore a asymp (ab)b hence b asymp (ab)a The proof ofthe second relation is similar

Corollary 2 In an associative magma (E middot) the inclusions

B sube (AB)A and B sube A(BA)

are valid for all nonvoid subsets A B of E

Definition 6 A hypercomposition (x y)rarr xperpy on a set E is called weakly associative if for allelements x y z in E

(xperpy)perpz asymp xperp(yperpz)

A magma whose law of synthesis is weakly associative is called a weakly associative magma

Example 11 Suppose that the law of synthesis on a magma E with more than three elements isthe following one

xperpy = E middot middotx y for every x y isin E with x 6= yxperpx = x for all x isin E

Then E is not an associative magma because

(xperpx)perpy = xperpy = E middot middotx y while xperp(xperpy) = xperp[E middot middotx y] = E middot middotx

However E is a weakly associative magma since

(xperpy)perpz cap xperp(yperpz) 6= empty f or all x y z in E

Proposition 5 The result of the hypercomposition in a weakly associative magma E is always anonempty set

Proof Suppose that xperpy = empty for some x y isin E Then (xperpy)perpz = empty for any z isin ETherefore (xperpy)perpz cap xperp(yperpz) = empty which is absurd Hence xperpy is nonvoid

Definition 7 A hypercomposition (x y)rarr xperpy on a set E is called weakly commutative if forall elements x y in E

xperpy asymp yperpx

A magma whose law of synthesis is weakly commutative is called a weakly commutative magma

Example 12 Let (E middot) be a magma and let be the extensive enlargement of the law of synthesisThen (E) is a weakly commutative magma since

x y sube x y cap y x

for all elements x y in E

Two statements of magma theory are dual statements if each one results from the otherby interchanging the order of the law of synthesis perp that is interchanging aperpb with bperpaObserve that the axiom of associativity is self-dual The two induced laws of synthesis perp

7


and perp have dual definitions hence they must be interchanged in a construction of a dualstatement Therefore the following principle of duality holds

Given a theorem the dual statement which results from the interchange of the order ofthe synthesis perp (and necessarily interchange perp and perp) is also a theorem

A direct consequence of the principle of duality is the following proposition

Proposition 6 The opposite law of an associative law of synthesis is associative

Proposition 7 The extensive enlargement of an associative law of synthesis is associative

Proof Let denote the extensive enlargement of an associative law of synthesis which iswritten multiplicatively then

(a b) c = [a b cup ab] c = a c cup b c cup (ab) c == a c cup ac cup b c cup bc cup ab cup c cup (ab)c == a b c cup ac cup bc cup ab cup a(bc) = a (b c)

Proposition 8 Let (Eperp) be an associative magma If an element a of E commutes with theelements b and c of E then it commutes with their synthesis as well

Proof aperp(bperpc) = (aperpb)perpc = (bperpa)perpc = bperp(aperpc) = bperp(cperpa) = (bperpc)perpa

Definition 8 A law of synthesis (x y)rarr xperpy on a set E is called reproductive if the equality

xperpE = Eperpx = E

is valid for all elements x in E A magma whose law of synthesis is reproductive is called areproductive magma

Example 13 The following laws of synthesis in E = 1 2 3 are reproductive

perp 1 2 3 perp 1 2 3

1 1 2 3 1 13 3 22 2 3 1 2 2 123 23 3 1 2 3 2 1 123

3 Groups and Hypergroups

Definition 9 An associative and reproductive magma is called a group if the law of synthesis on themagma is a composition while it is called a hypergroup if the law of synthesis is a hypercomposition

We observe that the two algebraic structures we just defined are equipped with thesame operating rules that is the axioms of associativity and reproductivity but withdifferent synthesizing laws for their elements Of course if the singletons are identifiedwith their members then the groups are special cases of the hypergroups The termgroup was first used in a technical sense by the French mathematician Eacutevariste Galois(25 October 1811ndash31 May 1832) His brilliant paper ldquoUne Meacutemoire sur les Conditions deReacutesolubiliteacute des Equations par Radicauxrdquo was submitted in January 1831 for the thirdtime to the French Academy of Sciences It was followed by his famous letter describinghis discoveries written the night before he was killed in a duel at the age of 21 in May1832 Galoisrsquos manuscripts with annotations by Joseph Liouville were published in 1846in the Journal de Matheacutematiques Pures et Appliqueacutees Eacutevariste Galois discovered the notion ofnormal subgroups and realized their importance Another young French mathematician

8


Freacutedeacuteric Marty (23 June 1911ndash14 June 1940) who was born a hundred years later whileworking on cosets determined by not normal subgroups introduced the hypergroup Hismathematical heritage on hypergroups was only three papers [1ndash3] as he was killed atthe age of 29 during World War II in the Gulf of Finland while serving in the French AirForce as a lieutenant For the record Freacutedeacuteric Martyrsquos father Joseph Marty was also amathematician who was killed in 1914 when Freacutedeacuteric was only 3 years old during WorldWar I

Example 14 Let E = a There is only one associative and reproductive magma whose law ofsynthesis is defined by the following table

perp a

a a

Example 15 Let E = a b There is only one group with two elements The following tableshows this group

perp a b

a a bb b a

On the other hand there exist eight nonisomorphic hypergroups with two elements The followingtables display these hypergroups Observe that the first one is the above group

perp a b perp a b perp a b

a a b a a b a a bb b a b b a b b a b a b


a a a b a a a b a a a bb b a b b a b b b a b a b

perp a b perp a b

a b a b a a b a bb a b a b b a b a b

Example 16 Let E = a b c There is only one group with three elements The following tabledisplays this group

perp a b c

a a b cb b c ac c a b

However there are 3999 nonisomorphic hypergroups with three elements [79] presents asoftware that can calculate and print all these 3999 hypergroups

Example 17 A magma endowed with the b-hypercomposition is a hypergroup called b-hypergroup

Example 18 A magma endowed with the total hypercomposition is a hypergroup called totalhypergroup

Theorem 1 Let (G middot) be a group or a hypergroup If ldquordquo is the extensive enlargement of ldquomiddotrdquothen (G) is a hypergroup

9


Proof By Proposition 7 the extensive enlargement of middot is associative Moreover

x H = H x = x cup H = H

and the theorem holds

4 The Reproductive Axiom in Groups

Recall the definition of the group that is mentioned in the previous section

Definition 10 (FIRST DEFINITION OF A GROUP) An associative and reproductive magma iscalled a group if the law of synthesis on the magma is a composition

In other words a group is a set of elements equipped with a law of composition thatis associative and reproductive The next theorems give some important properties ofthe group structure In this section unless otherwise indicated the law of synthesis is acomposition that will be written multiplicatively and G will denote a multiplicative group

Theorem 2 Let G be a group Then

i There exists an element e isin G such that ea = a = ae for all a isin Gii For each element a isin G there exists an element aprime isin G such that aprimea = e = aaprime

Proof (i) Let x isin G By reproductive axiom x isin xG Consequently there exists e isin Gfor which xe = x Next let y be an arbitrary element in G Since the composition isreproductive y isin Gx therefore there exists z isin G such that y = zx Consequentlyye = (zx)e = z(xe) = zx = y In an analogous way there exists an element e such thatey = y for all y isin G Then the equality e = ee = e is valid Therefore there exists e isin Gsuch that ea = a = ae for all a isin G

(ii) Let a isin G By reproductive axiom e isin a G Thus there exists aprime isin G such thate = aaprime Also by reproductive axiom e isin Ga Therefore there exists aprimeprime isin G such thate = aprimeprimea However

aprime = eaprime =(aprimeprimea)aprime = aprimeprime

(aaprime)= aprimeprimee = aprimeprime

Hence aprime and aprimeprime coincide Therefore aaprime = e = aprimea

The element e is called the neutral element of G or the identity of G Moreover aprime iscalled the symmetric of a in G If the composition is written multiplicatively then e is calledthe unit of G and it is denoted by 1 Furthermore aprime is called the inverse of a and it isdenoted by aminus1 If the composition is written additively then e is called the zero of G and itis denoted by 0 Also aprime is called the opposite of a and it is denoted by minusa

Corollary 3 For each a b isin G

ab = abminus1 and ba = bminus1a

Corollary 4 Let e be the identity of a group G Then

eb = bminus1 and be = bminus1

for all b isin G


bb = e = bb and (bb)b = bminus1 = b(bb)

10


for all b isin G

Theorem 3 Let G be an associative magma whose law of synthesis is a composition Then G is agroup if the following two postulates are fulfilled


Proof It must be proved that the reproductive axiom is valid for G Since aG sube G for alla isin G it has to be proved that G sube aG Suppose that x isin G Then

x = ex =(aaprime)x = a

(aprimex)

The product aprimex is an element of G thus x isin aG Hence G sube aG therefore aG = GSimilarly Ga = G

Theorems 2 and 3 lead to another definition of the group

Definition 11 (SECOND DEFINITION OF A GROUP) An associative magma G in which the lawof synthesis is a composition is called a group if


Yet one-half of the above definitionrsquos postulates (i) and (ii) can be omitted by thefollowing dual propositions

Proposition 9 The postulates (i) and (ii) of Definition 11 are equivalent to

i There exists an element e isin G with ea = a for all a isin Gii For each element a isin G there exists an element aprime isin G such that aprimea = e


i There exists an element e isin G with ae = a for all a isin Gii For each element a isin G there exists an element aprime isin G such that aaprime = e

We quote the following well known and important Theorems 4 5 6 7 8 and 9 whichcan be easily proved with the use of the second definition of the group because we want toshow in the next sections that similar theorems can be proved only in very specific types ofhypergroups

Theorem 4 The neutral element of a group is unique

Theorem 5 The symmetric of each element of a group is unique

Theorem 6 The symmetric of the neutral element is the neutral element itself

Theorem 7 For each a isin G (aminus1)minus1

= a

Theorem 8 For each a b isin G(ab)minus1 = bminus1aminus1

11


Theorem 9 The cancellation law

ac = bc implies a = bca = cb implies a = b

Theorem 10 A finite associative magma G is a group if the law of synthesis on the magma is acomposition in which the cancellation law holds

Proof Let G = a1 middot middot middot an and let a be an arbitrary element in G Then

aG = aa1 middot middot middot aan sube G

From the cancellation law it follows that the n elements of aG are all distinct ThereforeaG and G have the same cardinality Consequently since G is finite aG = G is valid Dualitygives Ga = G and so the theorem

Theorem 11 An associative magma G whose law of synthesis is a composition is a group if andonly if the inverse laws are compositions

Proof Let G be a group We will prove that ba and ab are elements of G for all pairsof elements a b of G By reproduction Ga = G for all a isin G Consequently for everyb isin G there exists x isin G such that b = xa Thus x = ba Dually ab is an element of GConversely now suppose that the right quotient ba exists for all pairs of elements a b ofG Thus for each a b isin G there is an element x in G such that b = xa Therefore G sube Gafor all a isin G Next since Ga sube G for all a isin G it follows that Ga = G for all a isin G In asimilar way the existence of the left quotient ab for all pairs of elements a b of G yieldsaG = G for all a isin G Thus the reproductive law is valid and so G is a group

Corollary 6 An associative magma G is a group if and only if the equations

xa = b and ay = b

are solvable for all pairs of elements a b of G

Proof By Theorem 11 G is a group if and only if ba and ab are elements of G for all a bin G which equivalently implies that the equations xa = b and ay = b respectively aresolvable for all pairs of elements a b of G

Having proved the above another definition can be given for the group

Definition 12 (THIRD DEFINITION OF A GROUP) An associative magma G in which the lawof synthesis is a composition is called a group if the right quotient ba and the left quotient abresult in a single element of G for all a b isin G

Or else

An associative magma G in which the law of synthesis is a composition is called a group ifthe equations

xa = b and ay = b


Definition 13 A group that has the additional property that for every pair of its elements

a b = b a

is called an Abelian (after NH Abel 1802-29) or commutative group

12


In abstract algebra we also consider structures that do not satisfy all the axioms ofa group

Definition 14 An associative magma in which the law of synthesis is a composition is called asemigroup A semigroup with an identity is called a monoid

Definition 15 A reproductive magma in which the law of synthesis is a composition is called aquasigroup A quasigroup with an identity is called loop

Definition 16 A magma that is the union of a group with an absorbing element is called almost-group

5 Fundamental Properties of Hypergroups

Recall the earlier mentioned definition for the hypergroup

Definition 17 (FIRST DEFINITION OF A HYPERGROUP) An associative and reproductivemagma is called a hypergroup if the law of synthesis on the magma is a hypercomposition

In this section unless otherwise indicated the law of synthesis is a hypercompositionthat will be written multiplicatively and H will denote a multiplicative hypergroup

Theorem 12 ab 6= empty is valid for all the elements a b of a hypergroup H

Proof Suppose that ab = empty for some a b isin H By the reproductive axiom aH = H andbH = H Hence

H = aH = a(bH) = (ab)H = emptyH = empty

which is absurd

Theorem 13 ab 6= empty and ba 6= empty is valid for all the elements a b of a reproductive magma E

Proof By the reproductive axiom Eb = E for all b isin E Hence for every a isin E there existsx isin E such that a isin xb Thus x isin ab and therefore ab 6= empty Dually ba 6= empty

Theorem 14 If ab 6= empty and ba 6= empty for all pairs of elements a b of a magma E then E is areproductive magma

Proof Suppose that xa 6= empty for all a x isin E Thus there exists y isin E such that x isin yaTherefore x isin Ea for all x isin E and so E sube Ea Next since Ea sube E for all a isin E it followsthat E = Ea By duality E = aE

Following Theorem 14 another definition of the hypergroup can be given

Definition 18 (SECOND DEFINITION OF A HYPERGROUP) An associative magma is called ahypergroup if the law of synthesis is a hypercomposition and the result of each one of the two inversehypercompositions is nonvoid for all pairs of elements of the magma

Theorem 15 In a hypergroup H the equalities

i H = Ha = aH andii H = aH = Haare valid for all a in H

Proof (i) By Theorem 12 the result of the hypercomposition in H is always a nonemptyset Thus for every x isin H there exists y isin H such that y isin xa which implies that x isin yaHence H sube Ha Moreover Ha sube H Therefore H = Ha Next let x isin H Since

13


H = xH there exists y isin H such that a isin xy which implies that x isin ay Hence H sube aHMoreover aH sube H Therefore H = aH (ii) follows by duality

Likewise to the groups certain axioms were removed from the hypergroup thusrevealing the following weaker structures

Definition 19 A magma in which the law of synthesis is a hypercomposition is called a hyper-goupoid if for every two of its elements a b it holds that ab 6= empty otherwise it is called a partialhypergroupoid

In the case of finite hypergroupoids the ratio of the number of hypergroups overthe number of hypergroupoids is exceptionally small For instance we come across one3-element hypergroup in every 1740 hypergroupoids of three elements [79]

Definition 20 An associative hypergoupoid is called a semihypergroup while a reproductivehypergoupoid is called a quasihypergroup

Definition 21 The magma which is the union of a hypergroup with an absorbing element is calledalmost-hypergroup

Definition 22 A reproductive magma in which the law of synthesis is weakly associative is calledHV-group [80]

Because of Proposition 5 the result of the hypercomposition in an HV-group isalways nonvoid

Many papers have been written on the construction of examples of the above algebraicstructures Among them are the papers by P Corsini [8182] P Corsini and V Leoreanu [83]B Davvaz and V Leoreanu [84] I Rosenberg [85] I Cristea et al [86ndash90] M De Salvo andG Lo Faro [9192] C Pelea and I Purdea [9394] CG Massouros and CG Tsitouras [9596]and S Hoskova-Mayarova and A Maturo [70] in which hypercompositional structuresdefined in terms of binary relations are presented and studied

It is worth mentioning that a generalization of the hypergroup is the fuzzy hypergroupwhich was studied by a multitude of researchers [97ndash127] An extensive bibliography onthis subject can be found in [124] It can be proved that similar fundamental propertiesas the aforementioned ones are valid in the fuzzy hypergroups as well [125126] Forinstance a b 6= 0H is valid for any pair of elements a b in a fuzzy hypergroup (H ) [125]Generalizations of the fuzzy hypergroups are the mimic fuzzy hypergroups [125126] andthe fuzzy multihypergroups [127]

6 Types of Hypergroups

The hypergroup being a very general structure was equipped with further axiomswhich are more or less powerful and lead to a significant number of special hypergroupsOne such important axiom is the transposition axiom Initially this axiom was introducedby W Prenowitz in a commutative hypergroup all the elements of which also satisfy theproperties aa = a and aa = a He named this hypergroup join space and used it in thestudy of geometry [19ndash24] The transposition axiom in a commutative hypergroup is

ab cap cd 6= empty implies ad cap bc 6= empty

A commutative hypergroup that satisfies the transposition axiom is called a joinhypergroup Later on J Jantosciak generalized the transposition axiom in an arbitraryhypergroup H as follows

ba cap cd 6= empty implies ad cap bc 6= empty for all a b c d isin H

14


A hypergroup equipped with the transposition axiom is called transposition hyper-group [128]

These hypergroups attracted the interest of a large number of researchers includingI Cristea et al [102107ndash112] P Corsini [100ndash103129130] V Leoreanu-Fortea [101103ndash105131132]I Rosenberg [85132] S Hoskova-Mayerova [133ndash137] J Chvalina [135ndash138]P Rackova [135136] ChG Massouros [139ndash150] GG Massouros [144ndash152] J Nieminen [153154]A Kehagias [115] R Ameri [117118155] M M Zahedi [117] and G Chowdhury [123]

Proposition 11 [150] The following are true in any transposition hypergroup

i a(bc) sube abc and (cb)a sube cbaii a(cb) sube abc and (bc)a sube cbaiii (ba)(cd) sube (bac)d = b(acd)iv (ba)(cd) sube (bad)c = b(adc)v (ba)(cd) sube (abc)d = a(bcd)

The hypergroups are much more general algebraic structures than the groups to theextent that a theorem similar to Theorem 2 cannot be proved for the hypergroups In fact ahypergroup does not necessarily have an identity element Moreover in hypergroups thereexist different types of identities [149150156] In general an element e of a hypergroup His called right identity if x isin x middot e for all x in H If x isin e middot x for all x in H then x is called leftidentity while x is called identity if it is both a right and a left identity ie if x isin xe cap exfor all x isin H If the equality e = ee is valid for an identity e then e is called idempotentidentity A hypergroup H is called semiregular if every x isin H has at least one right and oneleft identity An identity is called scalar if a = ae = ea for all a isin H while it is called strongif ae = ea sube e a for all a isin H More generally an element s isin H is called scalar if theresult of the hypercomposition of this element with any element in H is a singleton thatis if sa isin H and as isin H for all a isin H If only the first membership relation is valid thens is called left scalar while if only the second relation is valid then s is called right scalarWhen a scalar identity exists in H then it is unique but the strong identity is not necessarilyunique Both scalar and strong identities are idempotent identities

Remark 1 If a hypercomposition has a scalar identity e then it is neither open nor closed(containing) because e isin ex and x asymp ex

Proposition 12 If e is a strong identity in H and x 6= e then xe = ex = x

Proof Let t isin xe Then x isin te sube t e Since x 6= e it follows that t = x Thus xe = xSimilarly it can be proven that ex = x

Corollary 7 If e is a strong identity in H and X is a nonempty subset of H not containing ethen Xe = eX = X

Proposition 13 If e is a scalar identity in H then xe = ex = x

Corollary 8 If X is a non-empty subset of H and if e is a scalar identity in H then Xe = eX = X

Theorem 16 i If a hypergroup H contains a scalar element then it contains a scalar identity eas wellii The set U of the scalar elements of a hypergroup H is a group

Proof (i) Let s be a scalar element Then per reproductivity there exists an element e isin Hsuch that se = s Also because of the reproductive axiom any element y isin H can bewritten as y = xs x isin H Hence ye = (xs)e = x(se) = xs = y Similarly ey = y

15


(ii) Let s isin U Then per reproductivity there exists an element sprime isin H such that ssprime = eIf y isin H then because of the reproductive axiom y = xs x isin H Hence ysprime = (xs)sprime =x(ssprime) = xe = x and therefore sprime is a right scalar element Similarly there exists a left scalarelement sprime prime such that sprime primes = e But sprime is equal to sprime prime since sprime prime = sprime primee = sprime prime(ssprime) =

(sprime primes)

sprime =esprime = sprime Consequently U is a subgroup of the hypergroup H

The group of the scalars was named the nucleus of H by Wall [11]An element xprime is called right e-symmetric of x or right e-inverse in the multiplicative

case if there exists a right identity e 6= xprime such that e isin x middot xprime The definition of the lefte-symmetric or left e-inverse is analogous to the above while xprime is called the e-inverse ore-symmetric of x if it is both right and left inverse with regard to the same identity e If eis an identity in a multiplicative hypergroup H then the set of the left inverses of x isin Hwith regard to e will be denoted by Sel(x) while Ser(x) will denote the set of the rightinverses of x isin H with regard to e The intersection Sel(x) cap Ser(x) will be denoted bySe(x) A semiregular hypergroup H is called regular if it has at least one identity e and ifeach element has at least one right and one left e-inverse H is called strictly e-regular iffor the identity e the equality Sel(x) = Ser(x) is valid for all x isin H In a strictly e-regularhypergroup the inverses of x are denoted by Se(x) and when there is no likelihood ofconfusion e can be omitted and the notation S(x) is used for the inverses of x We say thatH has semistrict e-regular structure if Sel(x) asymp Ser(x) is valid for any x isin H Obviously inthe commutative hypergroups there exist only the strict e-regular structures

Proposition 14 If e is an identity in a hypergroup H then Sel(x) = (ex) middot middote and Ser(x) =(xe) middot middote

Corollary 9 If Sel(x) cap Ser(x) 6= empty then xe cap ex 6= empty

Definition 23 A regular hypergroup is called reversible if it satisfies the following conditions

i z isin xyrArr x isin zyprime for some yprime isin S(y)ii z isin xyrArr y isin xprimez for some xprime isin S(x)

The enrichment of a hypergroup with an identity creates different types of hyper-groups depending on the type of the identity

Proposition 15 If H is a transposition hypergroup with a scalar identity e then for any x in Hthe quotients ex and xe are singletons and equal to each other

Proof Obviously ee = ee = e Let x 6= e Because of reproduction there exist xprime and xprimeprime such that e isin xprimex and e isin xxprimeprime Thus x isin xprimee and x isin exprimeprime Hence xprimee asymp exprimeprime Thereforebecause of transposition exprimeprime asymp xprimee is valid Since e is a scalar identity the following is truexprimeprime = exprimeprime and xprimee = xprime Thus xprime = xprimeprime However xprime isin ex and xprimeprime isin xe Therefore exand xe are equal and since this argument applies to any yprime yprimeprime isin H such that e isin yprimex ande isin xyprimeprime it follows that ex and xe are singletons

Definition 24 A transposition hypergroup that has a scalar identity e is called quasicanonicalhypergroup [157158] or polygroup [159ndash161]

The connection of quasicanonical hypergroups with color schemes relation algebrasand finite permutation groups as well as with weak cogroups produces a lot of exam-ples of quasicanonical hypergroups (eg see [160]) In [162] quasicanonical hypergroupsappear as Pasch geometries In a Pasch geometry (A∆e) A becomes a quasicanoni-cal hypergroup with scalar identity e and aminus1 = a when ab =

x∣∣(a b x) isin ∆

(see

also [128]) The following example from [158] shows the structural relation of groups withthe quasicanonical hypergroups

16


Example 19 Let(G middot) be a group and e its identity The following hypercomposition is definedon G

x y = x y x middot y i f y 6= xminus1 x y 6= ex xminus1 = xminus1 x = G i f x 6= ex e = e x = x f or all x isin G

Then (G ) becomes a quasicanonical hypergroup Note that this construction can be used toproduce new quasicanonical hypergroups from other quasicanonical hypergroups

In the quasicanonical hypergroups there exist properties analogous to (i) and (ii) ofTheorem 2 which are valid in the groups

Theorem 17 [128141] If (Q middot e) is a quasicanonical hypergroup then

i For each x isin Q there exists one and only one xprime isin Q such that e isin xxprime = xprimexii z isin xyrArr x isin zyprime rArr y isin xprimez

Corollary 10 A quasicanonical hypergroup is a reversible hypergroup

The inverse of Theorem 17 is also true

Theorem 18 [128141] If a hypergroup Q has a scalar identity e and


then the transposition axiom is valid in Q

When the hypercomposition is written multiplicatively xprime is denoted by xminus1 and itis called the inverse of x while if the hypercomposition is written additively the identityis denoted by 0 and the unique element xprime is called opposite or negative and it is denotedby minusx

Proposition 16 In a quasicanonical hypergroup Q

i xminus1 = ex = xe and x = exminus1 = xminus1e for all x isin Qii xy = xyminus1 and yx = yminus1x for all x y isin Q

Proof (i) is a direct consequence of Proposition 15 Next for (ii) applying (i) Proposition11 and Corollary 8 we have

xy = x(

eyminus1)sube xyminus1e = xyminus1

and xyminus1 = x(ey) sube xey = xy

Therefore the first equality is proved The second one arises from duality

The aforementioned Theorems 4 5 6 7 and 8 which are valid in the groups are alsovalid in the quasicanonical hypergroups The cancellation law (Theorem 9) though isvalid in the quasicanonical hypergroups as follows

ac = bc implies bminus1a cap ccminus1 6= emptyca = cb implies baminus1 cap ccminus1 6= empty

More generally the following theorem is valid

Theorem 19 If Q is a quasicanonical hypergroup then ab asymp cd implies that bdminus1 asymp aminus1c andcminus1a asymp dbminus1

17


Proof As it is mentioned above Theorem 8 is valid in the quasicanonical hypergroups aswell Indeed the equality (ab)minus1 = bminus1aminus1 follows from the sequence of implications

x isin ab a isin xbminus1 aaminus1 sube xbminus1aminus1 e isin xbminus1aminus1 xminus1 isin bminus1aminus1

Next let x isin ab cap cd Then xxminus1 isin ab(cd)minus1 = abdminus1cminus1 Hence e isin abdminus1cminus1 thusc isin abdminus1 Therefore there exists y isin bdminus1 such that c isin ay Reversibility implies thaty isin aminus1c Consequently bdminus1 asymp aminus1c By duality cminus1a asymp dbminus1

A commutative quasicanonical hypergroup is called canonical hypergroup The canoni-cal hypergroup owes its name to J Mittas [163164] while it was first used by M Krasnerfor the construction of the hyperfield which is a hypercompositional structure that he in-troduced in order to define a certain approximation of complete valued fields by sequencesof such fields [53] The hyperfields which were constructed afterward contain interestingexamples of canonical hypergroups [165ndash170] An example of such a canonical hypergroupis presented in the following construction by J Mittas [171]

Example 20 Let E be a totally ordered set and 0 its minimum element The following hypercompo-sition is defined on E

x + y =

maxx y i f x 6= yz isin E | z le x i f x = y

Then (E+) is a canonical hypergroup

We cite the above example because the hyperfield which J Mittas constructed basedon this canonical hypergroup is now called a tropical hyperfield (see eg [54ndash58]) andit is used in the development of the tropical geometry Example 19 also gives a canonicalhypergroup when (G middot) is an abelian group The hyperfield that is constructed basedon this canonical hypergroup leads to open problems in both hyperfield and field theo-ries [168169]

J Mittas studied the canonical hypergroup in depth [163164172ndash180] Also motivatedby the valuated hyperfield theory he introduced ultrametric distances to the canonicalhypergroups thus defining the valuated and the hypervaluated canonical hypergroupsNext he proved that the necessary and sufficient condition for a canonical hypergroup tobe valuated or hypervaluated is the validity of certain additional properties of a purelyalgebraic type that is properties that are expressed without the intervention of the valua-tion or the hypervaluation respectively Thus three special canonical hypergroups cameinto being

(a) The strongly canonical hypergroup which also satisfies the axiomsS1 If x isin x + a then x + a = x for all x a isin HS2 If (x + y) cap (z + w) 6= empty then either x + y sube z + w or z + w sube x + y

(b) The almost strongly canonical hypergroup which also satisfies the above axiom S2 andthe axiom

AS If x 6= y then either (xminus x) cap (yminus x) = empty or (yminus y) cap (yminus x) = empty

(c) The superiorly canonical which is a strongly canonical hypergroup that also satisfiesthe axioms

S3 If z w isin x + y and x 6= y then zminus z = wminus wS4 If x isin zminus z and y isin zminus z then xminus x sube yminus y

J Mittas has presented a very deep and extensive study on these hypergroups with agreat number and variety of results among which we mention the following theorem [178]

Theorem 20 The necessary and sufficient condition for a canonical hypergroup to be hypervaluatedis to be superiorly canonical

18


In all the above cases the neutral element is scalar Let us now consider hypergroupsthat are equipped with a strong identity

Definition 25 A fortified transposition hypergroup (FTH) is a transposition hypergroup T with aunique strong identity e which satisfies the axiom

For every x isin T middot middote there exists one and only one element xprime isin T middot middote the symmetricof x such that e isin xxprime and furthermore for xprime it holds that e isin xprimex

If the hypercomposition is commutative the hypergroup is called a fortified join hypergroup (FJH)

The fortified join hypergroup was introduced for the study of languages and automatawith tools of hypercompositional algebra [35ndash43]

It has been proved that every fortified transposition hypergroup consists of twotypes of elements the canonical (c-elements) and the attractive (a-elements) [141144149] Anelement x is called canonical if ex = xe is the singleton x while it is called attractive ifex = xe = e x A denotes the set of the attractive elements (a-elements) and C denotesthe set of the canonical elements (c-elements) By convention e isin A

Proposition 17 If (T middot e) is a fortified transposition hypergroup then the following are valid

i If x 6= e then xe = ex = xii ee = ee = Aiii If a isin A then aa = aa = Aiv If x y are attractive elements then x y sube xyv If x y are attractive elements then x isin xy and x isin yxvi If a isin A and c isin C then ac = ca = cvii If a isin A and a 6= e then ea = eaminus1 = e a = aminus1e = aeviii If x y are attractive elements then xyminus1 = xy cup

yminus1 and yminus1x = yx cup

yminus1

A detailed and thorough study of the properties of the a-elements and c-elements ispresented in [141144149] Theorems 4 5 6 and 7 which are valid for the groups are alsovalid for the fortified transposition hypergroups (FTH) Theorem 8 though is not valid inthe FTH as generally

(aaminus1)minus1 6= aaminus1 (or minus(aminus a) 6= aminus a in the additive case) This led

to the definition of two types of elements those that satisfy the equality(aaminus1)minus1

= aaminus1

(or minus(aminus a) = aminus a in the additive case) which are called normal and for which Theorem8 is valid and the rest which are called abnormal [141144149]

Theorem 21 If (Q middot)is a quasicanonical hypergroup and ldquordquo is the extensive enlargement of ldquomiddotrdquothen(Q) is a fortified transposition hypergroup consisting of attractive elements only

Proof Per Proposition 7 the extensive enlargement of an associative law of synthesis isalso associative Next for the proof of the transposition axiom we observe that

ba cap cd = [ba cup a] cap [cd cup c] anda d cap b c = [ad cup a d] cap [bc cup b c]

Therefore if ba asymp cd we distinguish the casesif ba asymp cd then ad asymp bc thus a d asymp b cif c isin ba then a isin bc thus a isin a d cap b c consequently a d asymp b cif a isin cd then c isin ad thus c isin a d cap b c consequently a d asymp b cif a = c then a d asymp b c

19


Finally if e is the neutral element of the quasicanonical hypergroup then

a e = ae cup a e = a e for all a isin Q

Example 21 Let H be a totally ordered set dense and symmetric around a center denoted by 0 isin HThe partition H = H minus cup 0 cup H + is defined with regard to this center and according to itfor every x isin H minus and y isin H + it is x lt 0 lt y and for every x y isin H x le yrArr minusy le minusx where ndashx is the symmetric of x with regard to 0 Then H equipped with the hypercomposition

x + y = x y i f y 6= minusx

and x + (minusx) = [ 0 |x| ] cup minus|x|

becomes an FJH in which x minus x 6= minus(xminus x) for every x 6= 0 So all the elements of (H+)are abnormal

The fortified transposition hypergroup is closely related to the quasicanonical hyper-group as per the following structure theorem

Theorem 22 [141] A transposition hypergroup H containing a strong identity e is isomorphic tothe expansion of the quasicanonical hypergroup C cup e by the transposition hypergroup A of allattractive elements with regard to the identity e

Definition 26 A transposition polysymmetrical hypergroup (TPH) is a transposition hypergroup(P middot) with an idempotent identity e which satisfies the axioms

i x isin xe = exii For every x isin P middot middote there exists xprime isin P middot middote the symmetric of x such that e isin xxprime and

furthermore xprime satisfies e isin xprimex

The set of the symmetric elements of x is denoted by S(x) A commutative transposition polysym-metrical hypergroup is called a join polysymmetrical hypergroup (JPH)

A direct consequence of this definition is that for a nonidentity element x S(x)cupe =xe = ex when x is attractive and S(x) = xe = ex when x is non attractive

Example 22 Let K be a field and G a subgroup of its multiplicative group In K we define ahypercomposition ldquourdquo as follows

xu y = z isin K | z = xp + yq p q isin G

Then (Ku) is a join polysymmetrical hypergroup having the 0 of K as its neutral element Sincexu 0 = xq q isin G the neutral element 0 is neither scalar nor strong The symmetric set ofan element x of K is S(x) = minusxp | p isin G

Example 23 Let (Ai middot) i isin I be a family of fortified transposition hypergroups that consist onlyof attractive elements and suppose that the hypergroups Ai i isin I have a common identity e ThenT = cup

iisinIAi becomes a transposition polysymmetrical hypergroup under the hypercomposition

a b = ab if a b are elements of the same hypergroup Aia b = a e b if a isin Ai middot middote b isin Aj middot middote and i 6= j

Observe that e is a strong identity in T Moreover if a isin Ai then S(a) = (T middot middotAi) cup aprime wherearsquo is the inverse of a in Ai

20


Example 24 Let ∆i i isin I be a family of totally ordered sets that have a common minimum elemente The set ∆ = cup

iisinI∆i with hypercomposition

xy =

[minx y maxx y] i f x y isin ∆i i isin I

[e x] cup [ e y ] i f x isin ∆i y isin ∆j and i 6= j i j isin I

becomes a JPH with neutral element e

Proposition 18 If x is an attractive element of a transposition polysymmetrical hypergroup thenS(x) consists of attractive elements

Proof Let e isin ex Then x isin ee Moreover x isin exprime for any xprime isin S(x) Thusexprime cap ee 6= empty which by the transposition axiom gives ee cap exprime 6= empty or e isin exprime Hence xprime

is attractive

Proposition 19 The result of the hypercomposition of two attractive elements in a transpositionpolysymmetrical hypergroup consists of attractive elements only while the result of the hypercompo-sition of an attractive element with a non attractive element consists of non attractive elements

Proposition 20 If the identity in a transposition polysymmetrical hypergroup is strong thenx y sube xy and x isin xy x isin yx for any two attractive elements x y

The algebraic properties of transposition polysymmetrical hypergroups are studiedin [145146]

Definition 27 A quasicanonical polysymmetrical hypergroup is a hypergroup H with a uniquescalar identity e which satisfies the axioms

i For every x isin H there exists at least one element xprime isin H called symmetric of x such thate isin xxprime and e isin xprimex

ii If z isin xy there exist xprime yprime isin S(x) such that x isin zyprime and y isin xprimez

A commutative quasicanonical polysymmetrical hypergroup is called a canonical polysymmetricalhypergroup

Example 25 Let H be a set that is totally ordered and symmetric around a center denoted by0 isin H Then H equipped with the hypercomposition

x + y =

y i f |x| lt |y| f or every x y isin H minus cup 0 or x y isin 0 cup H +

[x y] i f x isin H minus and y isin H +

becomes a canonical polysymmetrical hypergroup Suppose now that x y a b isin H+ andx lt y lt a lt b Then xy = ab = H minus Thus xy cap ab 6= empty However x + b = band y + a = a Therefore (x + b) cap (y + a) = empty and so the transposition axiom is not valid

The canonical polysymmetrical hypergroup was introduced by J Mittas [181] Inaddition J Mittas and Ch Massouros while studying the applications of hypergroupsin the linear spaces defined the generalized canonical polysymmetrical hypergroup [31]Moreover J Mittas in his paper [174] motivated by an observation about algebraicallyclosed fields discovered a special type of completely regular polysymmetrical hypergroupwhich later on C Yatras called M-polysymmetrical hypergroup C Yatras in a series ofpapers studied this hypergroup and its properties in detail [182ndash184] J Mittas also definedthe generalized M-polysymmetrical hypergroups that were studied by himself and by ChMassouros [185186]

21


Definition 28 A M-polysymmetrical hypergroup H is a commutative hypergroup with an idem-potent identity e that also satisfies the axioms

i x isin xe = exii For every x isin H middot middote there exists at least one element xprime isin H middot middote such that e isin xxprime and

e isin xprimex (symmetric of x)iii If z isin xy and xprime isin S(x) yprime isin S(y) zprime isin S(z) then zprime isin xprime yprime

Proposition 21 [182183] Every M-polysymmetrical hypergroup is a join hypergroup

In addition to the aforementioned hypergroups other hypergroups have been definedand studied Among them are the complete hypergroups and the complete semihyper-groups [17102107112187ndash189] the 1-hypergroups [102107190] the hypergroups of typeU [191ndash198] the hypergroups of type C [199200] and the cambiste hypergroup [28]

7 Subhypergroups and Cosets

Decompositions and partitions play an important role in the study of algebraic struc-tures Undoubtedly this study is of particular interest in the theory of hypercompositionalalgebra It has recently been addressed in various papers from different perspectives(eg [188201ndash204]) Moreover in [33] it is proved that general decomposition theoremsthat are valid in hypergroups give as corollaries well-known decomposition theorems inconvex sets A far-reaching concept of abstract group theory is the decomposition of agroup into cosets by its subgroups The hypergroup though being a more general struc-ture than that of the group has various types of subhypergroups In contrast to groupswhere any subgroup decomposes the group into cosets in the hypergroups not all thesubhypergroups can define such a partition This section presents the subhypergroups thatcan create a partition in the hypergroup and the relevant partitions

Definition 29 A nonempty subset K of H is a semi-subhypergroup when it is stable under thehypercomposition ie it has the property xy sube K for all x y isin K K is a subhypergroup of H if itsatisfies the reproductive axiom ie if the equality xK = Kx = K is valid for all x isin K

71 Closed and Ultra-Closed Subhypergroups

From the above Definition 29 it derives that when K is a subhypergroup and a b isin Kthe relations a isin bx and a isin yb always have solutions in K If for any two elements a andb in K all the solutions of the relation a isin yb lie inside K then K is called right closed inH Similarly K is le f t closed when all the solutions of the relation a isin bx lie in K K isclosed when it is both right and left closed Note that the concepts subhypergroup andclosed subhypergroup are self-dual A direct consequence of the definition of the closedsubhypergroup is the proposition

Proposition 22 The nonvoid intersection of two closed subhypergroups is a closed subhypergroup

The relevant property is not valid for every subhypergroup since although thenonvoid intersection of two subhypergroups is stable under the hypercomposition thevalidity of the reproductive axiom fails This was one of the reasons that led from thevery beginning of the hypergroup theory to the consideration of more special types ofsubhypergroups one of which is the above defined closed subhypergroup (eg see [513])An equivalent definition of the closed hypergroup is the following one

Definition 30 [205206] A subhypergroup K of H is called right closed if K is stable under theright division ie if ab sube K for all a b isin K K is called left closed if K is stable under the leftdivision ie if ba sube K for all a b isin K K is called closed when it is both right and left closed

22


Proposition 23 If the hypercomposition in a hypergroup H is closed (containing) then H has noproper closed subhypergroups

Proof According to Proposition 1 if a hypercomposition is closed then aa = H for alla isin H Consequently the only closed subhypergroup of H is H itself

Proposition 24 If K is a subset of a hypergroup H such that ab sube K and ba sube K for alla b isin K then K is a closed subhypergroup of H

Proof Initially it will be proved that K is a hypergroup ie that aK = Ka = K for any a inK Let x isin K Then ax sube K Therefore x isin aK Hence K sube aK For the reverse inclusionnow suppose that y isin aK Then Ky sube KaK So K cap (KaK)y 6= empty Thus y isin (KaK)KPer mixed associativity the equality KaK = (KK)a is valid Thus

(KaK)K = ((KK)a)K sube (Ka)K sube (KK)K sube KK sube K

Hence y isin K and so aK sube K Therefore aK = K The equality Ka = K follows by dualityThe rest comes from Definition 30

Proposition 25 If K is a closed hypergroup of a hypergroup H then

K = Ka = aK = aK = Ka

for all a in K

Proposition 26 If K is a subhypergroup of H then

H middot middotK sube (H middot middotK)x and H middot middotK sube x(H middot middotK)for all x isin K

Proof Let y isin H middot middotK and y isin (H middot middotK)x Per reproductive axiom y isin Hx and sincey isin (H middot middotK)x y must be a member of Kx Thus y isin Kx sube KK = K This contradicts theassumption and so H middot middotK sube (H middot middotK)x The second inclusion follows by duality

Proposition 27 i A subhypergroup K of H is right closed in H if and only if (H middot middotK)x = H middot middotKfor all x isin Kii A subhypergroup K of H is left closed in H if and only if x(H middot middotK) = H middot middotK for all x isin Kiii A subhypergroup K of H is closed in H if and only if x(H middot middotK) = (H middot middotK)x = H middot middotK for allx isin K

Proof (i) Let K be right closed in H Suppose that z isin H middot middotK and zx cap K 6= empty Then thereexists an element y in K such that y isin zx or equivalently z isin yx Therefore z isin K whichis absurd Hence (H middot middotK)x sube H middot middotK Next because of Proposition 26 H middot middotK sube (H middot middotK)xand therefore H middot middotK = (H middot middotK)x Conversely now suppose that (H middot middotK)x = H middot middotK for allx isin K Then (H middot middotK)x cap K = empty for all x isin K Hence t isin rx and so r isin tx for all x t isin Kand r isin H middot middotK Therefore tx cap (H middot middotK) = empty which implies that tx sube K Thus K is rightclosed in H (ii) follows by duality and (iii) is an obvious consequence of (i) and (ii)

Corollary 11 i If K is a right closed subhypergroup in H then xK cap K = empty for all x isin H middot middotKii If K is a left closed subhypergroup in H then Kx cap K = empty for all x isin H middot middotKiii If K is a closed subhypergroup in H then xK cap K = empty and Kx cap K = empty for all x isin H middot middotK

23


Proposition 28 Let A be a nonempty subset of a hypergroup H and suppose that

A0 = A cup AA cup AA cup AAA1 = A0 cup A0 A0 cup A0A0 cup A0A0

An = Anminus1 cup Anminus1 Anminus1 cup Anminus1Anminus1 cup Anminus1Anminus1

Then 〈A〉 = cupnge0

An is the least closed subhypergroup of H which contains A

If A is a singleton then the above procedure constructs the monogene closed sub-hypergroup of the hypergroup H It is easy to see that if the hypercomposition is openthen 〈a〉 = a while if it is closed 〈a〉 = H for all a isin H The notion of the monogenesubhypergroups was introduced by J Mittas in [177] for the case of the canonical hyper-groups In [67] there is a detailed study of the monogene symmetric subhypergroups ofthe fortified join hypergroups These studies highlight the existence of two types of theorder of an element the principal order which is an integer and the associated order which isa function A subcategory of the monogene subhypergroups is the cyclic subhypergroupsthat is subhypergroups of the form

cupkisinN

ak a isin H

It is easy to observe that in the case of the open hypercompositions the cyclic subhy-pergroup which is generated by an element a is the element a itself The study of cyclicsubhypergroups has attracted the interest of many researchers [207ndash218] A detailed studyand thorough review of cyclic hypergroups is given in [219]

The closed subhypergroups are directly connected with the ultra-closed subhyper-groups [17220] The properties of these subhypergroups together with their ability tocreate cosets in special hypergroups were studied in a long series of joint papers by MDe Salvo D Freni D Fasino and G Lo Faro [191ndash195] D Freni [196ndash198] M Gutanet al [198ndash200] and L Haddad and Y Sureau [221222] A new definition for the ultra-closed subhypergroups with the use of the induced hypercompositions is given belowAfter that Theorem 24 proves that this definition is equivalent to Sureaursquos definition

Definition 31 A subhypergroup K of a hypergroup H is called right ultra-closed if it is closed andaa sube K for each a isin H K is called left ultra-closed if it is closed and aa sube K for each a isin H IfK is both right and left ultra-closed then it is called ultra-closed

Theorem 23 i If K is right ultra-closed in H then either ab sube K or ab cap K = empty for alla b isin H Moreover if ab sube K then ba sube Kii If K is left ultra-closed in H then either ba sube K or ba cap K = empty for all a b isin H Moreoverif ba sube K then ab sube K

Proof Suppose that abcapK 6= empty a b isin H Then a isin sb for some s isin K Next assume thatba cap (H middot middotK) 6= empty Then b isin ta t isin H middot middotK Thus a isin s(ta) = (st)a Since K is closed perProposition 27 st sube H middot middotK So a isin ra for some r isin H middot middotK Therefore aa cap (H middot middotK) 6= emptywhich is absurd Hence ba sube K Now let x be an element in K such that b isin xa Ifab cap (H middot middotK) 6= empty there exists y isin H middot middotK such that a isin yb Therefore b isin x(yb) = (xy)bSince K is closed per Proposition 27 xy sube H middot middotK So b isin zb for some z isin H middot middotK Thereforebb cap (H middot middotK) 6= empty which is absurd Hence ab sube K Duality gives (ii)

Theorem 24 Let H be a hypergroup and K a subhypergroup of H Theni K is a right ultra-closed subhypergroup of H if and only if Ka cap (H middot middotK)a = empty for all a isin Hii K is a left ultra-closed subhypergroup of H if and only if aK cap a(H middot middotK) = empty for all a isin H

24


Proof Suppose that K is a right ultra-closed subhypergroup of H Then aa sube K for alla isin H Since K is right closed then (aa)s sube K is valid or equivalently a(as) sube Kfor all s isin K Theorem 23 yields (as)a sube K for all s isin K If Ka cap (H middot middotK)a 6= empty thenthere exist s isin K and r isin H middot middotK such that sa cap ra 6= empty which implies that r isin asa But(as)a sube K hence r isin K which is absurd Conversely now let Ka cap (H middot middotK)a = empty for alla isin H If a isin K then K cap (H middot middotK)a = empty Therefore s isin ra for every s isin K and r isin H middot middotKEquivalently sacap (H middot middotK) = empty for all s isin K Hence sa sube K for all s isin K and a isin K So Kis right closed Next suppose that aa cap (H middot middotK) 6= empty for some a isin H Then a isin (H middot middotK)aor Ka sube K(H middot middotK)a Since K is closed per Proposition 27 K(H middot middotK) sube H middot middotK is valid ThusKa sube (H middot middotK)a which contradicts the assumption Duality gives (ii)

72 Invertible Subhypergroups and their Cosets

Definition 32 A subhypergroup K of a hypergroup H is called right invertible if ab cap K 6= emptyimplies ba cap K 6= empty with a b isin H while it is called left invertible if ba cap K 6= empty impliesab cap K 6= empty with a b isin H K is invertible when it is both right and left invertible

Note that the concept of the invertible subhypergroup is self-dual Theorem 4 in [134]gives an interesting example of an invertible subhypergroup in a join hypergroup of partialdifferential operators

Theorem 25 If K is invertible in H then K is closed in H

Proof Let x isin KK Then K asymp xK thus K asymp xK Since K is invertible K asymp xKimplies K asymp Kx from which it follows that x isin KK Since K is a subhypergroup KK = KTherefore x isin K Consequently KK sube K Since invertibility is self-dual KK sube K is validas well

The converse of Theorem 25 is not true In [17] there are examples of closed hyper-groups that are not invertible On the other hand the following proposition is a directconsequence of Theorem 23

Proposition 29 i The right (left) ultra-closed subhypergroups of a hypergroup are right (left) invertibleii The ultra-closed subhypergroups of a hypergroup are invertible

Theorem 25 and Proposition 23 give the following result

Proposition 30 A hypergroup H has no proper invertible subhypergroups if its hypercompositionis closed (containing)

Proposition 31 A hypergroup H has no proper invertible subhypergroups if the hypercompositionis open

Proof Suppose that K is an invertible subhypergroup of H and a is an element of H thatdoes not belong to K Because of the reproductive axiom there exists b isin H such thata isin Kb Therefore ab cap K 6= empty which per invertibility of K implies that ba cap K 6= emptyHence b isin Ka and so

a isin Kb sube K(Ka) = (KK)a = Ka

Thus aa cap K 6= empty But according to Proposition 1 aa = a Therefore a isin K whichcontradicts the assumption

Proposition 32 i K is right invertible in H if and only if the implication a isin Kb rArr b isin Kais valid for all a b isin Hii K is left invertible in H if and only if the implication b isin aK rArr a isin bK is valid for all a b isin H

25


Lemma 1 i If the implication Ka 6= KbrArr Ka cap Kb = empty is valid for all a b isin H then c isin Kcfor each c isin Hii If the implication aK 6= bK rArr aK cap bK = empty is valid for all a b isin H then c isin cK for eachc isin H

Proof (i) There exists x isin H such that c isin Kx Hence Kc sube Kx Therefore Kc cap Kx 6= emptySo c isin Kx = Kc (ii) is the dual of (i)

Theorem 26 i K is right invertible in H if and only if the implication Ka 6= KbrArr Ka cap Kb = emptyis valid for all a b isin Hii K is left invertible in H if and only if the implication aK 6= bK rArr aK cap bK = empty is valid forall a b isin H

Proof (i) Let K be right invertible in H Assume that Ka 6= Kb and that c isin Ka cap KbThen Kc sube Ka and Kc sube Kb Moreover c isin Ka cap Kb implies that ca cap K 6= empty andcb cap K 6= empty But K is invertible consequently we have the sequence of the followingequivalent statements

ac cap K 6= empty and bc cap K 6= empty a isin Kc and b isin Kc Ka sube Kc and Kb sube Kc

Therefore Ka = Kc and Kb = Kc Thus Ka = Kb which contradicts the assumptionSo Ka cap Kb = empty Conversely now suppose that Ka 6= KbrArr Ka cap Kb = empty is valid for alla b isin H and moreover assume that ab cap K 6= empty Then a isin Kb consequently Ka sube Kb Since KacapKb 6= empty it derives that Ka = Kb Per Lemma 1 b isin Kb Therefore b isin Ka whichimplies that ba cap K 6= empty (ii) is the dual of (i)

Theorem 27 i If K is right invertible in H then H K = Kx | x isin H is a partition in Hii If K is left invertible in H then H K = xK | x isin H is a partition in Hiii If K is invertible in H then H



Proposition 30 A hypergroup H has no proper invertible subhypergroups if its hypercomposi‐

tion is closed (containing)

Proposition 31 A hypergroup H has no proper invertible subhypergroups if the hypercomposi‐

tion is open

Proof Suppose that K is an invertible subhypergroup of H and a is an element of H

that does not belong to K Because of the reproductive axiom there exists b H such

that a Kb Therefore a b K which per invertibility of K implies that

b a K Hence b Ka and so

a Kb K Ka KK a Ka

Thus a a K But according to Proposition 1 a a a Therefore a K which

contradicts the assumption

Proposition 32 i Kis right invertible in H if and only if the implication a K b b K a is

valid for all a b H

ii Kis left invertible in H if and only if the implication b aK a bK is valid for all a b H

Lemma 1 i If the implication Ka Kb Ka Kb is valid for all a b H then c Kc

for each c H

ii If the implication aK bK aK bK is valid for all a b H then c cK for each

c H

Proof (i) There exists x H such that c Kx Hence Kc Kx Therefore

Kc Kx So c Kx Kc (ii) is the dual of (i)

Theorem 26 i Kis right invertible in H if and only if the implication Ka Kb Ka Kb

is valid for all a b H

ii Kis left invertible in H if and only if the implication aK bK aK bK is valid for all

a b H

Proof (i) Let K be right invertible in H Assume that Ka Kb and that

c Ka Kb Then Kc Ka and Kc Kb Moreover c Ka Kb implies that

c a K and c b K But K is invertible consequently we have the se‐

quence of the following equivalent statements

a c K and b c K a Kc and b Kc Ka Kc and Kb Kc

Therefore Ka Kc and Kb Kc Thus Ka Kb which contradicts the assumption

So Ka Kb Conversely now suppose that Ka Kb Ka Kb is valid for all a b H and moreover assume that a b K Then a K b consequently

Ka K b Since Ka Kb it derives that Ka Kb Per Lemma 1 b Kb Therefore

b Ka which implies that b a K (ii) is the dual of (i)

Theorem 27 i If Kis right invertible in H then |H K Kx x H is a partition in H

ii If Kis left invertible in H then |H K xK x H is a partition in H iii If Kis in‐

vertible in H then |H K KxK x H is a partition in H

Proof The proofs of (i) and (ii) are similar to the proof of (iii) So we prove (iii)

K = KxK | x isin H is a partition in H

Proof The proofs of (i) and (ii) are similar to the proof of (iii) So we prove (iii)(iii) According to Lemma 1 if c is an arbitrary element in H then c isin Kc and c isin cKc isin Kc implies cK sube KcK and since c isin cK it derives that c isin KcK Now suppose thatthere exist a b isin H such that KaK cap KbK 6= empty Let c isin KaK cap KbK c isin KaK implies thatthere exists y isin Ka such that c isin yK However due to the invertibility of K it follows thata isin Ky and y isin cK Hence a isin KcK and therefore KaK sube KcK Moreover c isin KaK impliesthat KcK sube KaK Consequently KaK = KcK In the same way KbK = KcK and thereforeKaK = KbK

Theorem 28 Let K be a right invertible subhypergroup of a hypergroup H and let X = Kx x isin HThen X lowastY = XY K is a hypercomposition in H K = Kx | x isin H and (H K lowast) isa hypergroup

Corollary 12 If K is a subgroup of a group G then (G K lowast) is a hypergroup

Theorem 29 Suppose that K is a subgroup of a group G and K sube M sube G Then (M K lowast) isa subhypergroup of (G K lowast) if and only if M is a subgroup of a group G

Proof Suppose that M K is a subhypergroup of G K Then

Kx lowast (M K) = M K = (M K) lowast Kx for all Kx isin M K

26


Therefore KxM = M = MKx for all x isin M Since e isin K it follows that xM = M andMx = M for all x isin M Thus M is a subgroup of G Conversely now let M be a subgroupof G and x isin M Then

Kx lowast (M K) = KxM K = KM K = M K

and ( M K ) lowast Kx = MKx K = Mx K = M K

Consequently (M K lowast) is a subhypergroup of (G K lowast)

73 Reflexive Closed Subhypergroups and their Cosets

As shown above when the hypercomposition is open or closed there do not existproper invertible subhypergroups So in such cases the hypergroup cannot be decomposedinto cosets with the use of the previous techniques and different methods need to bedeveloped in order to solve the decomposition problem Such a method that uses aspecial type of closed subhypergroups is presented below for the case of the transpositionhypergroups

Definition 33 A subhypergroup N of a hypergroup H is called normal or invariant if aN = Nafor all a isin H

Proposition 33 N is an invariant subhypergroup of a hypergroup H if and only if Na = aNfor all a isin H

Proof Suppose that N is invariant in H Then we have the sequence of equivalentstatements

x isin Na a isin Nx a isin xN x isin aN

Therefore Na = aN Conversely now suppose that Na = aN Then we have thefollowing equivalent statements

x isin aN a isin xN a isin Nx x isin Na

Consequently N is invariant

Definition 34 A subhypergroup R of a hypergroup H is called reflexive if aR = Ra for alla isin H

Obviously all the subhypergroups of the commutative hypergroups are invariant andreflexive The closed and reflexive subhypergroups have a special interest in transpositionhypergroups because they can create cosets

Theorem 30 If L is a closed and reflexive subhypergroup of a transposition hypergroup T thenthe sets Lx form a partition in T

Proof Let Lx cap Ly 6= empty Then xL cap Ly 6= empty and therefore xL cap Ly 6= empty Con-sequently x isin LyL Next using the properties mentioned in Propositions 11 and 3successively we have

Lx sube L(LyL) sube LLLy = LLy = (Ly)L = L(Ly) = (LL)y = Ly

Hence Lx sube Ly By symmetry Ly sube Lx and so Lx = Ly

According to the above theorem for each x isin T there exists a unique class thatcontains x This unique class is denoted by Lx The set of the classes modulo L is denotedby TL Note that L isin TL Indeed since L is closed Lx = L for any x isin L

27


Proposition 34 If xy cap L 6= empty then Lx = Ly and Ly = Lx

Proof xy cap L 6= empty implies both x isin Ly and y isin xL = Lx Therefore Lx = Ly andLy = Lx

From the above proposition it becomes evident that Lx and Lx are different classesin TL The following theorems reveal the algebraic form of the class Lx x isin T

Theorem 31 If Lx is a class then LLx is also a class modulo L and LLx = Ly for somey isin Lx

Proof Let y isin Lx It suffices to prove that LLx = Ly because this implies that LLxis a class modulo L Since y isin Lx it follows that Ly sube LLx For the proof of thereverse inclusion let z isin LLx Then L cap zLx 6= empty hence Lx cap zL = Lx cap Lz 6= emptyBy Theorem 30 Lx = Lz and so y isin Lz Then yz cap L 6= empty and z isin yL = Ly Consequently LLx sube Ly hence the theorem

Theorem 32 If Lx is the class modulo L of an element x in T then

Lx = LxL = LLx = L(Lx) = (xL)L

Proof Let x isin T Since L is reflexive xL = Lx Applying the transposition axiom weget xL asymp Lx which implies that x isin LxL and x isin LxL Next from Proposition 11 (ii)it follows that L(Lx) sube LxL For the proof of the reverse inclusion Corollary 2 andProposition 11 (ii) sequentially give

LxL sube [L(LxL)]L sube (LLLx)L = (LLx)L

Since L is reflexive the equality (LLx)L = L(LLx) holds Therefore Proposition 3and Proposition 11 (ii) give

L(LLx) = L[(Lx)L] sube LL(Lx) = L(Lx)

Consequently L(Lx) = LxL Duality yields the rest

Corollary 13 If T is a quasicanonical hypergroup then Lx = xL

Proof Per Proposition 16 xL = xLminus1 thus

Lx = (xL)L =(

xLminus1)

Lminus1 = (xL)L = x(LL) = xL

Proposition 35 If L is a reflexive closed subhypergroup of a transposition hypergroup T then

(Lx)(Ly) sube Lxy x y isin T

Proof The successive application of Proposition 11 (iii) and mixed associativity gives

(Lx)(Ly) = (xL)(Ly) sube (xLL)y = (xL)y = (Lx)y = Lxy

A direct consequence of the above proposition is that the partition which is defined byL is regular Therefore a hypercomposition ldquo

Mathematics 2021 9 0 28 of 40

By Theorem 30 Lx = Lz and so y isin Lz Then yz cap L = empty and z isin yL = Ly Consequently LLx sube Ly hence the theorem








Corollary 13 If Tis a quasicanonical hypergroup thenLx = xL


Lx = (xL)L =(

xLminus1)






A direct consequence of the above proposition is that the partition that is defined by Lis regular Therefore a hypercomposition middot is defined in TL by

(Lx) middot (Ly) = Lz | z isin xy

Next it is apparent that L is the neutral element of the above hypercomposition andthat the transposition is valid Hence the theorem

Theorem 33 If L is a reflexive closed subhypergroup of a transposition hypergroup T then(TL middot L) is a quasicanonical hypergroup

Example 26 A three-dimensional Euclidian space becomes a hypergroup under the hypercomposi-tion defined in Example 4 It is easy to verify that this is a commutative transposition hypergroup ofidempotent elements that is a join space The notation E3

J is used for this hypergroup A line L ofthe Euclidian space is a reflexive closed subhypergroup of E3

J The cosets that L defines in E3J are the

halfplanes Lx which are drawn in the following Figure 4(

E3J

)L

is a quasicanonical hypergroup

L is its neutral element and the symmetric of any element Lx is the element L(Lx)

add a space before the wordis and after the word then

n that i

replace the word that withthe word which

middot

replace the dot

with a thicker dot in quotationmarks mid hight of the letters

middot

middot

replace the dot

with a thicker dot mid hight of theletters

rdquo is defined in TL by

(Lx)












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


(Ly) = Lz | z isin xy


28


Theorem 33 If L is a reflexive closed subhypergroup of a transposition hypergroup T then(TL












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


L) is a quasicanonical hypergroup





E3J

)L


L is its neutral element and the symmetric of any element Lx is the element L(Lx)Mathematics 2021 9 x FOR PEER REVIEW 32 of 45

Figure 4 Example 26

74 Symmetric Subhypergroups and Their Cosets

As mentioned above when the hypercomposition in a hypergroup is closed (contain‐

ing) then there do not exist closed subhypergroups In this case other types of subhyper‐

groups must be used for the decomposition of the hypergroup An example of such a

hypergroup is the transposition hypergroup with idempotent identity which consists of

attractive elements only As shown in the sixth section of this paper the transposition

hypergroups can have one (fortified transposition hypergroups) or more (transposition

polysymmetrical hypergroups) symmetric elements for each one of their elements Obvi‐

ously a transposition polysymmetrical hypergroup becomes a fortified transposition hy‐

pergroup when the set S(x) of the symmetric elements of any one of its elements x is a

singleton and moreover when its identity is strong The quite general case of the decom‐

position of the transposition polysymmetrical hypergroups with strong identity into co‐

sets is pesented below Since the result of the hypercomposition between two attractive

elements contains these two elements there do not exist proper closed subhypergroups

There exist though subhypergroups that contain the symmetric element of each one of

their elements These subhypergroups are the ones that decompose the transposition pol‐

ysymmetrical hypergroups into cosets In the following P signifies a transposition poly‐symmetrical hypergroup with strong identity

Definition 35 A subhypergroup M of a transposition polysymmetrical hypergroup is called

symmetric if S x M for all x in M

Proposition 36 Let x y be two elements in P such that e xS y then

xS y x y S y and S y x y x S y

Proof xS y x e y xe y x e y x y e y x y e S y Since e

is not in xS y it follows that xS y x y S y Moreover since e yS y it fol‐

lows that y e S y Therefore x y x e S y Hence

x y S y x e S y S y xS y e S y xS y S y xS y

Figure 4 Example 26

74 Symmetric Subhypergroups and their Cosets

As mentioned above when the hypercomposition in a hypergroup is closed (contain-ing) then there do not exist closed subhypergroups In this case other types of subhy-pergroups must be used for the decomposition of the hypergroup An example of sucha hypergroup is the transposition hypergroup with idempotent identity which consistsof attractive elements only As shown in the sixth section of this paper the transpositionhypergroups can have one (fortified transposition hypergroups) or more (transpositionpolysymmetrical hypergroups) symmetric elements for each one of their elements Ob-viously a transposition polysymmetrical hypergroup becomes a fortified transpositionhypergroup when the set S(x) of the symmetric elements of any one of its elements x is asingleton and moreover when its identity is strong The quite general case of the decompo-sition of the transposition polysymmetrical hypergroups with strong identity into cosets ispesented below Since the result of the hypercomposition between two attractive elementscontains these two elements there do not exist proper closed subhypergroups There existthough subhypergroups that contain the symmetric element of each one of their elementsThese subhypergroups are the ones that decompose the transposition polysymmetricalhypergroups into cosets In the following P signifies a transposition polysymmetricalhypergroup with strong identity

Definition 35 A subhypergroup M of a transposition polysymmetrical hypergroup is calledsymmetric if S(x) sube M for all x in M

Proposition 36 Let x y be two elements in P such that e isin xS(y) then xS(y) = xy cup S(y)and S(y)x = yx cup S(y)

29


Proof xS(y) sube x(ey) sube xey = x ey = xy cup ey = xy cup e cup S(y) Since e is notin xS(y) it follows that xS(y) sube xy cup S(y) Moreover since e isin yS(y) it follows thaty isin eS(y) Therefore xy sube x(eS(y)) Hence

xy cup S(y) sube x(eS(y)) cup S(y) sube xS(y)e cup S(y) = xS(y) cup S(y) = xS(y)

Thus xS(y) = xy cup S(y) Dually S(y)x = yx cup S(y)

Proposition 37 Let M be a symmetric subhypergroup of P If x isin M then

i xM capM = empty and Mx capM = emptyii xM = xM cupM and Mx = Mx cupMiii Mx = M S(x) and xM = S(x)Miv (xM)M = xM and M(Mx) = Mx

Proof The two statements in each one of (i)ndash(iv) are dual and therefore it is sufficient toprove only one of them(i) Let x isin M and y isin M such that xy capM 6= empty Then x isin My = M which contradictsthe assumption for x Thus xM capM = empty(ii) Since M is symmetric S(M) = M Thus e isin xM since x isin M So via Proposition 36

xM = xS(M) = xM cup S(M) = xM cupM

(iii) According to Proposition 36 the equality M S(x) = Mx cup S(x) holds therefore

Mx = Mx cup ex = Mx cup S(x) = M S(x)

(iv) Since x isin xM it follows that xM sube (xM)M In addition because of (ii) xM subexM thus

(xM)M sube (xM)M = x(MM) = xM

Theorem 34 Let M be a symmetric subhypergroup of P If x y isin M then

i xM asymp yM implies xM = yMii Mx asymp My implies Mx = Myiii M(xM) asymp M(yM) implies M(xM) = M(yM)iv (Mx)M asymp (My)M implies (Mx)M = (My)M

Proof (i) xM cap yM 6= empty implies that x isin (yM)M Since y isin M from (iv) and (ii) ofProposition 37 it follows that (yM)M = yM sube yM cupM Thus x isin yM cupM Sincex isin M it follows that x isin yM Thus xM sube (yM)M = y(MM) = yM Bysymmetry yM sube xM Hence xM = yM(ii) is the dual of (i)(iii) Per Corollary 1 mixed associativity and Propositions 11 and 37 (iv) we have

M(xM) asymp M(yM)rArr (Mx)M asymp M(yM)rArr Mx asymp [M(yM)]MrArr

rArr Mx asymp M[(yM)M]rArr Mx asymp MyMrArr x isin yMrArr y isin xMrArr

rArr yM sube (xM)MrArr yM sube x(MM)rArr yM sube xMrArr M(yM) sube M(xM)

By symmetry M(xM) sube M(yM) thus the equality is validFinally (iv) is true because (Mx)M = M(xM) and (My)M = M(yM)

30


If x isin P and M is a nonempty symmetric subhypergroup of P then xlarrM

(ie the leftcoset of M determined by x) and dually xrarr

M(ie the right coset of M determined by x) are

given by

xlarrM

=

M i f x isin M

xM i f x isin Mand xrarr

M=

M i f x isin M

Mx i f x isin M

Since per Corollary 1 the equality (BA)C = B(AC) is valid in any hypergroupthe double coset of M determined by x can be defined by

xM =

M i f x isin MM(xM) = (Mx)M i f x isin M

Per Theorem 34 the distinct left cosets and right cosets as well as the double cosetsare disjoint Thus each one of the families

P M =

xlarrM| x isin P

P M =

xrarr

M| x isin P

and P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M = xM | x isin P

forms a partition of P If M is normal then it follows that xlarrM

= xrarrM

= xM Therefore

P M = P M = P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M

Proposition 38 Let M be a symmetric subhypergroup of P Then

i x isin xlarrM

x isin xrarrM

and x isin xM

ii xlarrMsube xM and xrarr

Msube xM

iii xM =(

xlarrM

)rarrM

=(

xrarrM

)larrM


xM = xM cupM and Mx = Mx cupM

Proof The two equalities are dual Per Proposition 36 the equality xS(M) = xM cup S(M)is valid But M is symmetric so S(M) = M Therefore xM = xM cupM


i xlarrM

M = xM = xlarrMcupM

ii MxrarrM

= Mx = xrarrMcupM

Proof (i) If x isin M the equality is true because each one of its parts equals to M If x isin Mthen the sequential application of Propositions 37 (iv) and 39 gives

xlarrM

M = (xM)M = xM = xM cupM = xlarrMcupM

Duality gives (ii)

Corollary 14 Let Q be a nonvoid subset of P and M a symmetric subhypergroup of P Then

QlarrM

M = QM = QlarrM

M cupM and MQrarrM

= MQ = QrarrMcupM


MxM = MxlarrM

= xM cupM = MxM = xrarrM

M = xM M

31


Proof Per Proposition 40 (i) MxM = M(

xlarrMcupM

)= Mxlarr

McupMM = Mxlarr

McupM and

since the hypercomposition is closed MxlarrMcup M = Mxlarr

M Per duality MxM = xrarr

MM

Next per Propositions 38 (iii) and 40 we have

MxM = M(

xlarrM

)rarrM

= MxlarrM

=(

xlarrM

)rarrMcupM = xM cupM

Duality yields the rest


(xy)larrM

M sube xlarrM

ylarrMcupM and (xy)rarr

MM sube xrarr

Myrarr

McupM

Proof Per Corollary 14 (xy)larrMsube (xy)larr

MM = (xy)M But x isin xM thus we have

(xy)M sube (xM)yM = xlarrM(yM) Now per Proposition 40

xlarrM(yM) = xlarr

M

(yrarr

McupM

)= xlarr

Myrarr

Mcup xlarr

MM = xlarr

Myrarr

Mcup(

xlarrMcupM

)=(

xlarrM

yrarrMcup xlarr

M

)cupM

Since the hypercomposition is closed xlarrM

yrarrMcup xlarr

M= xlarr

Myrarr

Mis valid and therefore

(xlarr

Myrarr

Mcup xlarr

M

)cupM = xlarr

Myrarr

McupM

Duality gives the second inclusion

Corollary 15 If X Y are nonvoid subsets of P and M is a symmetric subhypergroup of P then

(XY)larrM

M sube XlarrM

YlarrMcupM and (XY)rarr

MM sube Xrarr

MYrarr

McupM


(xy)M = xMyM cupM

Proof Per Proposition 38 (iii) and Corollary 15

(xy)M =((xy)larr

M

)rarrMsube(

xlarrM

ylarrMcupM

)rarrM

=(

xlarrM

ylarrM

)rarrMcupMrarr

Msube

sube(

xlarrM

)rarrM

(xlarr

M

)rarrMcupM = xMyM cupM


(XY)M = XMYM cupM


i XMYM capM 6= empty implies (XMYM)M = XMYM cupMii XMYM capM = empty implies (XMYM)M = XMYM

A hypercomposition that derives from Prsquos hypercomposition can be defined in eachone of the families P M P M and P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M Thus in P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M this hypercomposition is

xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM = zM | z isin xMyM

32


If the induced hypercompositions of












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


are denoted by


M M M M M M M M M M

M M M M M M M

x y z x y M z x y z Mz

x y z z M x y z M

Since M Mx y M and M M M M Mx y x y z it follows that M M M MM x y z is

valid Therefore

M M M M M M M M M M M MM M M MMx y z x y z M x y z M x y z

Duality yields M M M M M MM M Mx y z x y z and so the associativity is valid

Proposition 46 The transposition axiom is valid in P M

Proof Suppose that M M M My x z w Then

| |M M M M M M M Mp x y p q z q w

| |M M M M M Mp x y p q z q w

Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y

Propositions 44 45 and 46 give the theorem

Theorem 35 If M is a symmetric subhypergroup of P then P M is a transposition hy‐

pergroup

A consequence of Proposition 41 is that M M MM x x M x M for every Mx in

P M Hence

Proposition 47 M is a strong identity in the hypergroup P M

Proposition 48 P M consists only of attractive elements

Since M M M Mx y x y for all M Mx y P M the following is true

Proposition 49 The hypercomposition in P M is closed

Proposition 50 If y S x then M MM x y for all Mx P M

Propositions 47 and 50 give as a consequence the following theorem

Theorem 36 If M is a symmetric subhypergroup of P then P M is a transposition

polysymmetrical hypergroup

Corollary 18 If M is a symmetric subhypergroup of a fortified transposition hypergroup of at‐

tractive elements T then T M is a fortified transposition hypergroup M is its strong

identity and each one of its elements is attractive

75 The Cosets in Quasicanonical Hypergroups

and


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













then

xM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













yM = zM | xM isin zM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM = zM | x isin zMyM

and yM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













xM = zM | xM isin yM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


zM = zM | x isin yMzM

It is obvious that xM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













yM 6= empty and yM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













xM 6= empty Therefore according to Theorem14 the following proposition is valid

Proposition 44 The reproductive axiom is valid in(

P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)

In the families P M and P M the associativity may fail However it is valid inP






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M as per the next Proposition

Proposition 45 The associative axiom is valid in(

P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)

Proof It must be proved that

(xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM)












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


zM = xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


(yM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


zM)

This is true if and only if ((xMyM)MzM)M = (xM(yMzM)M)M So if xMyM cap M = emptythen Corollary 17 (ii) yields (xMyM)M = xMyM and the above equality is obvious IfxMyM capM 6= empty then Corollary 17 (i) yields (xMyM)M = xMyM cupM Hence

(xMyM)zM = (xMyM cupM)zM = xMyMzM cupMzM= xMyMzM cup zM cupM = xMyMzM cupM

Since xMyM cap M 6= empty and xMyM sube xMyMzM it follows that M sube (xMyMzM)M isvalid Therefore

((xMyM)MzM)M = (xMyMzM cupM)M = (xMyMzM)M cupM = (xMyMzM)M

Duality yields (xMyMzM)M = (xM(yMzM)M)M and so the associativity is valid

Proposition 46 The transposition axiom is valid in(

P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)

Proof Suppose that yM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













xM cap zM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













wM 6= empty Then

pM | xM isin yM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


pM cap qM | zM isin qM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


wM 6= emptyhArrhArr pM | x isin yM pM cap qM | z isin qMwM 6= empty

Therefore yprimex cap zwprime 6= empty for some yprime isin yM and wprime isin wM Thus xwprime cap yprimez 6= empty and so

xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


wM cap zM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM 6= empty


Theorem 35 If M is a symmetric subhypergroup of P then(

P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is a transposition

hypergroup

A consequence of Proposition 41 is that M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


xM = xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


M = xM M for every xM

in P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M Hence

Proposition 47 M is a strong identity in the hypergroup P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M

33


Proposition 48 P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M consists only of attractive elements

Since xM yM sube xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM for all xM yM isin P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M the following is true

Proposition 49 The hypercomposition in(

P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is closed

Proposition 50 If y isin S(x) then M isin xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM for all xM isin P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M



P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is a transposition


Corollary 18 If M is a symmetric subhypergroup of a fortified transposition hypergroup ofattractive elements T then

(T






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is a fortified transposition hypergroup M is its strong



It is apparent that the subgroups of a group are symmetric closed and invertible Thesame is true in the case of the quasicanonical hypergroups Indeed per Proposition 16xy = xyminus1 and yx = yminus1x therefore a subhypergroup U of a quasicanonical hypergroupQ is symmetric if and only if it is closed Moreover since the reversibility is valid inthe quasicanonical hypergroups if U is a symmetric subhypergroup of a quasicanonicalhypergroup Q then the implications a isin rbrArr b isin rminus1a and a isin br rArr b isin arminus1 hold forevery a b isin Q r isin U Therefore because of Proposition 32 U is invertible Hence theleft and right cosets of a subhypergroup U of a quasicanonical hypergroup Q are of theform aU and Ua a isin Q respectively and create partitions in Q The quotient hypergroupQ U defined by the left cosets and the quotient hypergroup Q U defined by theright cosets are transposition hypergroups but not necessarily quasicanonical ones Forexample although U is a right scalar identity for the left cosets since aU middot U = aU it neednot be a left scalar identity as well since aU asymp U middot aU

Double cosets have the form UaU and also create a partition in Q It is easy to observethat (UaU)minus1 = Uaminus1U and that U is a bilateral scalar identity Hence

Theorem 37 If U is a symmetric subhypergroup of a quasicanonical hypergroup Q then thequotient hypergroup of the double cosets Q






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














U is a quasicanonical hypergroup with scalaridentity U

Corollary 19 If U is a normal symmetric subhypergroup of a quasicanonical hypergroup Q thenthe left right and double cosets coincide and the quotient hypergroup Q






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














U is a quasicanonicalhypergroup with U as its scalar identity

8 Conclusions and Open Problems

This paper presents the structural relation between the hypergroup and the groupAs it has been proved these two algebraic structures satisfy the exact same axioms andtheir only difference appears in the law of synthesis of their elements The law of syn-thesis in the hypergroups is so general that it allowed this structurersquos enrichment withfurther axioms which created a significant number of special hypergroups Many of thesespecial hypergroups are presented in this paper along with examples their fundamentalproperties and the different types of their subhypergroups and applications Among

34


them the transposition hypergroup has important applications in geometry and computerscience [67] This hypergroup satisfies the following axiom


which was named transposition axiom [24] If this axiomrsquos implication is reversed thefollowing new axiom is created

ad cap bc 6= empty implies ba cap cd 6= empty for all a b c d isin H

We will call this axiom rev-transposition axiom (ie reverse transposition axiom) and therelevant hypergroup rev-transposition hypergroup (ie reverse transposition hypergroup) or rev-join hypergroup (ie reverse join hypergroup) in the commutative case The relation betweenthese two hypergroups is very interesting For example the following proposition is validin the rev-transposition hypergroups

Proposition 51 The following are true in any rev-transposition hypergroup

i abc sube a(bc) and cba sube (cb)aii abc sube a(cb) and cba sube (bc)aiii (bac)d = b(acd) sube (ba)(cd)iv (bad)c = b(adc) sube (ba)(cd)v (abc)d = a(bcd) sube (ba)(cd)

The comparison of Proposition 51 with Proposition 11 reveals that the reversal of theimplication in the transposition axiom reverses the inclusion relations in properties (i)ndash(v)The study of this hypergroup and its potential applications present an interesting openproblem of the theory of the hypercompositional algebra

However the following property also applies

ba cap cd 6= empty hArr ad cap bc 6= empty for all a b c d isin H

We will name this axiom bilateral transposition axiom Under this axiom the inclusionrelations of Propositions 11 and 51 become equalities Examples of hypergroups verify-ing the bilateral transposition axiom are the quasicanonical hypergroups the canonicalhypergroups and of course the groups and the abelian groups So the following questionarises Do there exist other hypergroups satisfying the bilateral transposition axiom apart from thequasicanonical and the canonical ones

In addition as it is shown in Section 7 many questions remain open in the decom-position of hypergroups Addressing these quesions necessitates the introduction of newtools and techniques as hypercompositional algebra is an off-the-map region of abstractalgebra which requires great caution in the application of processes for achieving correctresults It is very easy to be led to wrong conclusions when relying on methods and resultsof classical algebra Such a case is indicated in remark 2 in [67]

Author Contributions Both authors contributed equally to this work All authors have read andagreed to the published version of the manuscript

Funding This research received no external funding The APC was funded by the MPDIjournal Mathematics

Institutional Review Board Statement Not applicable

Informed Consent Statement Not applicable

Data Availability Statement Not applicable

Acknowledgments We wish to extend our sincere thanks to I Cristea S Hoskova-Mayerova andM Novaacutek for having urged us to write this paper during the round tables of the 1st Symposium

35


on Hypercompositional AlgebramdashNew Developments and Applications Thessaloniki Greece 2017ICNAAM 2017

Conflicts of Interest The authors declare no conflict of interest

References1 Marty F Sur une Geacuteneacuteralisation de la Notion de Groupe Huitiegraveme Congregraves des Matheacutematiciens Scand Stockholm 1934 45ndash492 Marty F Rocircle de la notion de hypergroupe dans lrsquo eacutetude de groupes non abeacuteliens C R Acad Sci (Paris) 1935 201 636ndash6383 Marty F Sur les groupes et hypergroupes attacheacutes agrave une fraction rationelle Ann Lrsquo Ecole Norm 1936 3 83ndash1234 Krasner M Sur la primitiviteacute des corps B-adiques Mathematica 1937 13 72ndash1915 Krasner M La loi de JordanmdashHolder dans les hypergroupes et les suites generatrices des corps de nombres Pmdashadiqes (I) Duke

Math J 1940 6 120ndash140 (II) Duke Math J 1940 7 121ndash135 [CrossRef]6 Krasner M La caracteacuterisation des hypergroupes de classes et le problegraveme de Schreier dans ces hypergroupes C R Acad Sci

(Paris) 1941 212 948ndash9507 Krasner M Hypergroupes moduliformes et extramoduliformes Acad Sci (Paris) 1944 219 473ndash4768 Krasner M Kuntzmann J Remarques sur les hypergroupes CR Acad Sci (Paris) 1947 224 525ndash5279 Kuntzmann J Opeacuterations multiformes Hypergroupes C R Acad Sci (Paris) 1937 204 1787ndash178810 Kuntzmann J Homomorphie entre systeacutemes multiformes C R Acad Sci (Paris) 1937 205 208ndash21011 Wall HS Hypergroups Am J Math 1937 59 77ndash98 [CrossRef]12 Ore O Structures and group theory I Duke Math J 1937 3 149ndash174 [CrossRef]13 Dresher M Ore O Theory of multigroups Am J Math 1938 60 705ndash733 [CrossRef]14 Eaton EJ Ore O Remarks on multigroups Am J Math 1940 62 67ndash71 [CrossRef]15 Eaton EJ Associative Multiplicative Systems Am J Math 1940 62 222ndash232 [CrossRef]16 Griffiths LW On hypergroups multigroups and product systems Am J Math 1938 60 345ndash354 [CrossRef]17 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Tricesimo Udine Italy 199318 Bourbaki N Eacuteleacutements de Matheacutematique Algegravebre Hermann Paris France 197119 Prenowitz W Projective Geometries as multigroups Am J Math 1943 65 235ndash256 [CrossRef]20 Prenowitz W Descriptive Geometries as multigroups Trans Am Math Soc 1946 59 333ndash380 [CrossRef]21 Prenowitz W Spherical Geometries and mutigroups Can J Math 1950 2 100ndash119 [CrossRef]22 Prenowitz W A Contemporary Approach to Classical Geometry Am Math Month 1961 68 1ndash67 [CrossRef]23 Prenowitz W Jantosciak J Geometries and Join Spaces J Reine Angew Math 1972 257 100ndash12824 Prenowitz W Jantosciak J Join Geometries A Theory of Convex Sets and Linear Geometry Springer BerlinHeidelberg

Germany 197925 Jantosciak J Classical geometries as hypergroups In Proceedings of the Atti del Convegno su Ipergruppi altre Structure

Multivoche et loro Applicazioni Udine Italy 15ndash18 October 1985 pp 93ndash10426 Jantosciak J A brif survey of the theory of join spaces In Proceedings of the 5th Intern Congress on Algebraic Hyperstructures

and Applications Iasi Romania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA 1994 pp 109ndash12227 Barlotti A Strambach K Multigroups and the foundations of Geometry Rend Circ Mat Palermo 1991 XL 5ndash68 [CrossRef]28 Freni D Sur les hypergroupes cambistes Rend Ist Lomb 1985 119 175ndash18629 Freni D Sur la theacuteorie de la dimension dans les hypergroupes Acta Univ Carol Math Phys 1986 27 67ndash8030 Massouros CG Hypergroups and convexity Riv Mat Pura Appl 1989 4 7ndash2631 Mittas J Massouros CG Hypergroups defined from linear spaces Bull Greek Math Soc 1989 30 63ndash7832 Massouros CG Hypergroups and Geometry Mem Acad Romana Math Spec Issue 1996 XIX 185ndash19133 Massouros CG On connections between vector spaces and hypercompositional structures Ital J Pure Appl Math 2015

34 133ndash15034 Dramalidis A On geometrical hyperstructures of finite order Ratio Math 2011 21 43ndash5835 Massouros GG Mittas J LanguagesmdashAutomata and hypercompositional structures In Proceedings of the 5th Intern Congress

on Algebraic Hyperstructures and Applications Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991 pp 137ndash14736 Massouros GG Automata Languages and Hypercompositional Structures PhD Thesis National Technical University of

Athens Athens Greece 199337 Massouros GG Automata and hypermoduloids In Proceedings of the 5th Intern Congress on Algebraic Hyperstructures and

Applications Iasi Romania 4ndash10 July 1993 Hadronic Press Palm Harbor FA USA 1994 pp 251ndash26538 Massouros GG An automaton during its operation In Proceedings of the 5th Internation Congress on Algebraic Hyperstructures

and Applications Iasi Romania 4ndash10 July 1993 Hadronic Press Palm Harbor FA USA 1994 pp 267ndash27639 Massouros GG On the attached hypergroups of the order of an automaton J Discret Math Sci Cryptogr 2003 6

207ndash215 [CrossRef]40 Massouros GG Hypercompositional structures in the theory of languages and automata An Sti Univ Al I Cuza Iasi Sect

Inform 1994 III 65ndash7341 Massouros GG Hypercompositional structures from the computer theory Ratio Math 1999 13 37ndash42

36


42 Massouros CG Massouros GG Hypergroups associated with graphs and automata AIP Conf Proc 2009 1168164ndash167 [CrossRef]

43 Massouros CG On path hypercompositions in graphs and automata MATEC Web Conf 2016 41 5003 [CrossRef]44 Chvalina J Chvalinova L State hypergroups of Automata Acta Math Inform Univ Ostrav 1996 4 105ndash12045 Chvalina J Krehliacutek S Novaacutek M Cartesian composition and the problem of generalizing the MAC condition to quasi-

multiautomata An St Univ Ovidius Constanta 2016 24 79ndash100 [CrossRef]46 Chvalina J Novaacutek M Krehliacutek S Hyperstructure generalizations of quasi-automata induced by modelling functions and signal

processing AIP Conf Proc 2019 2116 310006 [CrossRef]47 Novaacutek M Krehliacutek S Stanek D n-ary Cartesian composition of automata Soft Comput 2019 24 1837ndash1849 [CrossRef]48 Novaacutek M Some remarks on constructions of strongly connected multiautomata with the input semihypergroup being a

centralizer of certain transformation operators J Appl Math 2008 I 65ndash7249 Chvalina J Novaacutek M Smetana B Stanek D Sequences of Groups Hypergroups and Automata of Linear Ordinary Differential

Operators Mathematics 2021 9 319 [CrossRef]50 Krehliacutek S n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting

Mathematics 2020 8 835 [CrossRef]51 Chorani M Zahedi MM Some hypergroups induced by tree automata Aust J Basic Appl Sci 2012 6 680ndash69252 Chorani M State hyperstructures of tree automata based on lattice-valued logic RAIRO Theor Inf Appl 2018 52 23ndash4253 Krasner M Approximation des Corps Valueacutes Complets de Caracteacuteristique p 6=0 par Ceux de Caracteacuteristique 0 Colloque drsquo

Algegravebre Supeacuterieure (Bruxelles Decembre 1956) Centre Belge de Recherches Matheacutematiques Eacutetablissements Ceuterick LouvainLibrairie Gauthier-Villars Paris 1957 129ndash206

54 Viro O On basic concepts of tropical geometry Proc Steklov Inst Math 2011 273 252ndash282 [CrossRef]55 Viro O Hyperfields for Tropical Geometry I Hyperfields and dequantization arxiv 2010 arXiv1006303456 Baker M Bowler N Matroids over hyperfields arxiv 2017 arXiv16010120457 Jun J Geometry of hyperfields arxiv 2017 arXiv17070934858 Lorscheid O Tropical geometry over the tropical hyperfield arxiv 2019 arXiv19070103759 Connes A Consani C The hyperring of adegravele classes J Number Theory 2011 131 159ndash194 [CrossRef]60 Connes A Consani C From monoids to hyperstructures In search of an absolute arithmetic arXiv 2010 arXiv1006481061 Massouros CG Free and cyclic hypermodules Ann Mat Pura Appl 1988 150 153ndash166 [CrossRef]62 Mittas J Espaces vectoriels sur un hypercorps Introduction des hyperspaces affines et Euclidiens Math Balk 1975 5 199ndash21163 Vahedi V Jafarpour M Aghabozorgi H Cristea I Extension of elliptic curves on Krasner hyperfields Comm Algebra 2019 47

4806ndash4823 [CrossRef]64 Vahedi V Jafarpour M Cristea I Hyperhomographies on Krasner Hyperfields Symmetry 2019 11 1442 [CrossRef]65 Vahedi V Jafarpour M Hoskova-Mayerova S Aghabozorgi H Leoreanu-Fotea V Bekesiene S Derived Hyperstructures

from Hyperconics Mathematics 2020 8 429 [CrossRef]66 Freni D Strongly Transitive Geometric Spaces Applications to Hypergroups and Semigroups Theory Commun Algebra 2004 32

969ndash988 [CrossRef]67 Massouros GG Massouros CG Hypercompositional algebra computer science and geometry Mathematics 2020 8

1338 [CrossRef]68 Corsini P Leoreanu V Applications of Hyperstructures Theory Kluwer Academic Publishers Dordrecht The Netherlands 200369 Hoskova-Mayerova S Maturo A Decision-making process using hyperstructures and fuzzy structures in social sciences In Soft

Computing Applications for Group Decision-Making and Consensus Modeling Studies in Fuzziness and Soft Computing 357 CollanM Kacprzyk J Eds Springer International Publishing Cham Switzerland 2018 pp 103ndash111

70 Hoskova-Mayerova S Maturo A Algebraic hyperstructures and social relations Ital J Pure Appl Math 2018 39 701ndash70971 Hoskova-Mayerova S Maturo A Fuzzy Sets and Algebraic Hyperoperations to Model Interpersonal Relations In Recent Trends

in Social Systems Quantitative Theories and Quantitative Models Maturo A Hoskova-Mayerova S Soitu DT Kacprzyk J EdsStudies in Systems Decision and Control 66 Springer International Publishing Cham Switzerland 2017 pp 211ndash221

72 Hoskova-Mayerova S Maturo A An analysis of social relations and social group behaviors with fuzzy sets and hyperstructuresInt J Algebraic Hyperstruct Appl 2015 2 91ndash99

73 Hoskova-Mayerova S Maturo A Hyperstructures in social sciences AWER Procedia Inf Technol Comput Sci 2013 3 547ndash55274 Golzio AC Non-Deterministic Matrices Theory and Applications to Algebraic Semantics PhD Thesis IFCH University of

Campinas Campinas Brazil 201775 Golzio AC A brief historical Survey on hyperstructures in Algebra and Logic S Am J Log 2018 4 91ndash11976 Massouros CG Massouros GG On open and closed hypercompositions AIP Conf Proc 2017 1978 340002-1ndash340002-4 [CrossRef]77 Massouros CG Dramalidis A Transposition Hv-groups ARS Comb 2012 106 143ndash16078 Novaacutek M Krehliacutek Š EL-hyperstructures revisited Soft Comput 2018 22 7269ndash7280 [CrossRef]79 Tsitouras CG Massouros CG On enumeration of hypergroups of order 3 Comput Math Appl 2010 59 519ndash523 [CrossRef]80 Vougiouklis T Hyperstructures and Their Representations Hadronic Press Palm Harbor FL USA 199481 Corsini P Binary relations and hypergroupoids Ital J Pure Appl Math 2000 7 11ndash1882 Corsini P On the hypergroups associated with a binary relation Multi Val Log 2000 5 407ndash419

37


83 Corsini P Leoreanu V Hypergroups and binary relations Algebra Univers 2000 43 321ndash330 [CrossRef]84 Davvaz B Leoreanu-Fotea V Binary relations on ternary semihypergroups Comm Algebra 2010 38 3621ndash3636 [CrossRef]85 Rosenberg IG Hypergroups and join spaces determined by relations Ital J Pure Appl Math 1998 4 93ndash10186 Cristea I Several aspects on the hypergroups associated with n-ary relations An Stiint Univ Ovidius Constanta Ser Mat 2009

17 99ndash11087 Cristea I Stefanescu M Binary relations and reduced hypergroups Discret Math 2008 308 3537ndash3544 [CrossRef]88 Kankaras M Cristea I Fuzzy reduced hypergroups Mathematics 2020 8 263 [CrossRef]89 Cristea I Stefanescu M Angheluta C About the fundamental relations defined on the hypergroupoids associated with binary

relations Electron J Combin 2011 32 72ndash81 [CrossRef]90 Cristea I Jafarpour M Mousavi SS Soleymani A Enumeration of Rosenberg hypergroups Comput Math Appl 2011 32

72ndash81 [CrossRef]91 DeSalvo M LoFaro G Hypergroups and binary relations Multi Val Log 2002 8 645ndash65792 DeSalvo M LoFaro G A new class of hypergroupoids associated to binary relations J Mult Val Log Soft Comput 2003

9 361ndash37593 Pelea C Purdea I Direct limits of direct systems of hypergroupoids associated to binary relations in Advances in Mathematics

of Uncertainty In Proceedings of the Symposium ldquoMathematics of Uncertaintyrdquo ECIT 2006 Lasi Romania 20ndash23 September2006 pp 31ndash42

94 Breaz S Pelea C Purdea I Products of hypergroupoids associated to binary relations Carpathian J Math 2009 25 23ndash3695 Massouros CG Tsitouras CG Enumeration of hypercompositional structures defined by binary relations Ital J Pure Appl

Math 2011 28 43ndash5496 Tsitouras CG Massouros CG Enumeration of Rosenberg type hypercompositional structures defined by binary relations Eur

J Comb 2012 33 1777ndash1786 [CrossRef]97 Corsini P Tofan I On fuzzy hypergroups Pure Math Appl 1997 8 29ndash3798 Tofan I Volf AC On some connections between hyperstructures and fuzzy sets Ital J Pure Appl Math 2000 7 63ndash6899 Tofan I Volf AC Algebraic aspects of information organization In Systematic Organization of Information in Fuzzy Systems

Melo-Pinto P Teodorescu HN Fukuda T Eds 184 of NATO Science Series III Computer and Systems Sciences IOS PressAmsterdam The Netherlands 2003 pp 71ndash87

100 Corsini P Join Spaces Power Sets Fuzzy Sets In Proceedings of the 5th International Congress on Algebraic Hyperstructuresand Applications Iasi Romania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA 1994 pp 45ndash52

101 Corsini P Leoreanu V Fuzzy sets and join spaces associated with rough sets Rend Circ Mat Palermo 2002 51527ndash536 [CrossRef]

102 Corsini P Cristea I Fuzzy sets and non complete 1-hypergroups An Stalele Univ Ovidius Constanta 2005 13 27ndash54103 Corsini P Leoreanu-Fotea V On the grade of a sequence of fuzzy sets and join spaces determined by a hypergraph Southeast

Asian Bull Math 2010 34 231ndash242104 Leoreanu V Direct limit and inverse limit of join spaces associated with fuzzy sets Pure Math Appl 2000 11 509ndash516105 Leoreanu-Fotea V Fuzzy join n-ary spaces and fuzzy canonical n-ary hypergroups Fuzzy Set Syst 2010 161 3166ndash3173 [CrossRef]106 Leoreanu V About hyperstructures associated with fuzzy sets of type 2 Ital J Pure Appl Math 2005 17 127ndash136107 Cristea I Complete hypergroups 1-hypergroups and fuzzy sets An St Univ Ovidius Constanta 2000 10 25ndash38108 Cristea I A property of the connection between fuzzy sets and hypergroupoids Ital J Pure Appl Math 2007 21 73ndash82109 Stefanescu M Cristea I On the fuzzy grade of hypergroups Fuzzy Set Syst 2008 159 1097ndash1106 [CrossRef]110 Cristea I Davvaz B Atanassovrsquos intuitionistic fuzzy grade hypergroups Inf Sci 2010 180 1506ndash1517 [CrossRef]111 Davvaz B Sadrabadi EH Cristea I Atanassovrsquos intuitionistic fuzzy grade of ips hypergroups of order less than or equal to 6

Iran J Fuzzy Syst 2012 9 71ndash97112 Angheluta C Cristea I On Atanassovrsquos intuitionistic fuzzy grade of complete hypergroups J Mult Val Log Soft Comput 2013

20 55ndash74113 Cristea I Hyperstructures and fuzzy sets endowed with two membership functions Fuzzy Sets Syst 2009 160

1114ndash1124 [CrossRef]114 Cristea I Hoskova S Fuzzy Pseudotopological Hypergroupoids Iran J Fuzzy Syst 2009 6 11ndash19115 Kehagias A An example of L-fuzzy Join Space Rend Circ Mat Palermo 2002 51 503ndash526 [CrossRef]116 Kehagias A Lattice-fuzzy Meet and Join Hyperoperations In Proceedings of the 8th Intern Congress on Algebraic Hyperstruc-

tures and Applications Samothraki Greece 1ndash9 September 2002 Spanidis Press Xanthi Greece 2003 pp 171ndash182117 Ameri R Zahedi MM Hypergroup and join space induced by a fuzzy subset Pure Math Appl 1997 8 155ndash168118 Ameri R Fuzzy transposition hypergroups Ital J Pure Appl Math 2005 18 147ndash154119 Zahedi MM Bolurian M Hasankhani A On polygroups and fuzzy sub-polygroups J Fuzzy Math 1995 3 1ndash15120 Zahedi MM Hasankhani A F-Polygroups Int J Fuzzy Math 1996 4 533ndash548121 Zahedi MM Hasankhani A F-Polygroups (II) Inf Sci 1996 89 225ndash243 [CrossRef]122 Hasankhani A Zahedi MM Fuzzy sub-F-polygroups Ratio Math 1997 12 35ndash44123 Chowdhury G Fuzzy transposition hypergroups Iran J Fuzzy Syst 2009 6 37ndash52124 Davvaz B Cristea I Fuzzy Algebraic Hyperstructures Springer BerlinHeidelberg Germany 2015

38


125 Massouros CG Massouros GG On certain fundamental properties of hypergroups and fuzzy hypergroupsmdashMimic fuzzyhypergroups Intern J Risk Theory 2012 2 71ndash82

126 Massouros CG Massouros GG On 2-element Fuzzy and Mimic Fuzzy Hypergroups AIP Conf Proc 2012 14792213ndash2216 [CrossRef]

127 Hoskova-Mayerova S Al Tahan M Davvaz B Fuzzy multi-hypergroups Mathematics 2020 8 244 [CrossRef]128 Jantosciak J Transposition hypergroups Noncommutative Join Spaces J Algebra 1997 187 97ndash119 [CrossRef]129 Corsini P Graphs and Join Spaces J Comb Inf Syst Sci 1991 16 313ndash318130 Corsini P Binary relations interval structures and join spaces J Appl Math Comput 2002 10 209ndash216 [CrossRef]131 Leoreanu V On the heart of join spaces and of regular hypergroups Riv Mat Pura Appl 1995 17 133ndash142132 Leoreanu-Fotea V Rosenberg IG Join Spaces Determined by Lattices J Mult Val Log Soft Comput 2010 16 7ndash16133 Hoskova S Abelization of join spaces of affine transformations of ordered field with proximity Appl Gen Topol 2005 6

57ndash65 [CrossRef]134 Chvalina J Hoskova S Modelling of join spaces with proximities by first-order linear partial differential operators Ital J Pure

Appl Math 2007 21 177ndash190135 Hoskova S Chvalina J Rackova P Transposition hypergroups of Fredholm integral operators and related hyperstructures

(Part I) J Basic Sci 2006 3 11ndash17136 Hoskova S Chvalina J Rackova P Transposition hypergroups of Fredholm integral operators and related hyperstructures

(Part II) J Basic Sci 2008 4 55ndash60137 Chvalina J Hoskova-Mayerova S On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear

First-Order Partial Differential Operators An Stalele Univ Ovidius Constanta Ser Mat 2014 22 85ndash103 [CrossRef]138 Chvalina J Chvalinova L Transposition hypergroups formed by transformation operators on rings of differentiable functions

Ital J Pure Appl Math 2004 15 93ndash106139 Massouros CG Canonical and Join Hypergroups An Sti Univ Al I Cuza Iasi 1996 42 175ndash186140 Massouros CG Normal homomorphisms of fortified join hypergroups In Proceedings of the 5th Intern Congress on Algebraic

Hyperstructures and Applications Iasi Rommania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA 1994 pp 133ndash142141 Jantosciak J Massouros CG Strong Identities and fortification in Transposition hypergroups J Discret Math Sci Cryptogr

2003 6 169ndash193 [CrossRef]142 Massouros CG Isomorphism theorems in fortified transposition hypergroups AIP Conf Proc 2013 1558 2059ndash2062 [CrossRef]143 Massouros CG On the enumeration of rigid hypercompositional structures AIP Conf Proc 2015 1648 740005-1ndash

740005-6 [CrossRef]144 Massouros GG Massouros CG Mittas J Fortified join hypergroups Ann Math Blaise Pascal 1996 3 155ndash169 [CrossRef]145 Massouros GG Zafiropoulos FA Massouros CG Transposition polysymmetrical hypergroups In Proceedings of the 8th

Intern Congress on Algebraic Hyperstructures and Applications Samothraki Greece 1ndash9 September 2002 Spanidis PressXanthi Greece 2003 pp 191ndash202

146 Massouros CG Massouros GG Transposition polysymmetrical hypergroups with strong identity J Basic Sci 2008 4 85ndash93147 Massouros CG Massouros GG The transposition axiom in hypercompositional structures Ratio Math 2011 21 75ndash90148 Massouros CG Massouros GG On subhypergroups of fortified transposition hypergroups AIP Conf Proc 2013 1558

2055ndash2058 [CrossRef]149 Massouros CG Massouros GG On the algebraic structure of transposition hypergroups with idempotent identity Iran J

Math Sci Inform 2013 8 57ndash74150 Massouros CG Massouros GG Transposition hypergroups with idempotent identity Int J Algebr Hyperstruct Appl 2014

1 15ndash27151 Massouros GG Fortified join hypergroups and join hyperrings An Sti Univ Al I Cuza Iasi Sect I Mat 1995 XLI 37ndash44152 Massouros GG The subhypergroups of the fortified join hypergroup Ital J Pure Appl Math 1997 2 51ndash63153 Nieminen J Join Space Graphs J Geom 1988 33 99ndash103 [CrossRef]154 Nieminen J Chordal Graphs and Join Spaces J Geom 1989 34 146ndash151 [CrossRef]155 Ameri R Topological transposition hypergroups Ital J Pure Appl Math 2003 13 181ndash186156 Massouros CG Massouros GG Identities in multivalued algebraic structures AIP Conf Proc 2010 1281 2065ndash2068 [CrossRef]157 Bonansinga P Sugli ipergruppi quasicanonici Atti Soc Pelor Sc Fis Mat Nat 1981 27 9ndash17158 Massouros CG Quasicanonical hypergroups In Proceedings of the 4th Internation Congress on Algebraic Hyperstructures and

Applications Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991 pp 129ndash136159 Ioulidis S Polygroups et certaines de leurs proprieacutetieacutes Bull Greek Math Soc 1981 22 95ndash104160 Comer S Polygroups derived from cogroups J Algebra 1984 89 397ndash405 [CrossRef]161 Davvaz B Polygroup Theory and Related Systems World Scientific Singapore 2013162 Harrison DK Double coset and orbit spaces Pacific J Math 1979 80 451ndash491 [CrossRef]163 Mittas J Sur une classe drsquohypergroupes commutatifs C R Acad Sci Paris 1969 269 485ndash488164 Mittas J Hypergroupes values et hypergroupes fortement canoniques Proc Acad Athens 1969 44 304ndash312165 Krasner M A class of hyperrings and hyperfields Int J Math Math Sci 1983 6 307ndash312 [CrossRef]166 Massouros CG Methods of constructing hyperfields Intern J Math Math Sci 1985 8 725ndash728 [CrossRef]

39


167 Massouros CG On the theory of hyperrings and hyperfields Algebra Log 1985 24 728ndash742 [CrossRef]168 Massouros CG A class of hyperfields and a problem in the theory of fields Math Montisnigri 1993 1 73ndash84169 Massouros CG A Field Theory Problem Relating to Questions in Hyperfield Theory AIP Conf Proc 2011 1389

1852ndash1855 [CrossRef]170 Nakassis A Recent results in hyperring and hyperfield theory Intern J Math Math Sci 1988 11 209ndash220 [CrossRef]171 Mittas J Sur les hyperanneaux et les hypercorps Math Balk 1973 3 368ndash382172 Mittas J Hypergroupes canoniques hypervalues C R Acad Sci Paris 1970 271 4ndash7173 Mittas J Les hypervaluations strictes des hypergroupes canoniques C R Acad Sci Paris 1970 271 69ndash72174 Mittas J Hypergroupes et hyperanneaux polysymetriques C R Acad Sc Paris 1970 271 920ndash923175 Mittas J Contributions a la theacuteorie des hypergroupes hyperanneaux et les hypercorps hypervalues C R Acad Sc Paris 1971

272 3ndash6176 Mittas J Hypergroupes canoniques values et hypervalues Math Balk 1971 1 181ndash185177 Mittas J Hypergroupes canoniques Math Balk 1972 2 165ndash179178 Mittas J Contributions a la theacuteorie des structures ordonneacutees et des structures valueacutees Proc Acad Athens 1973 48 319ndash331179 Mittas J Hypergroupes canoniques values et hypervalues Hypergroupes fortement et superieurement canoniques Bull Greek

Math Soc 1982 23 55ndash88180 Mittas J Certaines remarques sur les hypergroupes canoniques hypervaluables et fortement canoniques Riv Mat Pura Appl

1991 9 61ndash67181 Mittas J Hypergroupes polysymetriques canoniques In Proceedings of the Atti del Convegno su Ipergruppi Altre Strutture

Multivoche e loro Applicazioni Udine Italy 15ndash18 October 1985 pp 1ndash25182 Yatras C M-polysymmetrical hypergroups Riv Mat Pura Appl 1992 11 81ndash92183 Yatras C Subhypergroups of M-polysymmetrical hypergroups In Proceedings of the 5th Intern Congress on Algebraic

Hyperstructures and Applications Iasi Rommania 4ndash10 July 1993 Hadronic Press Palm Harbor FA USA 1994 pp 123ndash132184 Yatras C Types of Polysymmetrical Hypergroups AIP Conf Proc 2018 1978 340004-1ndash340004-5 [CrossRef]185 Mittas J Generalized M-Polysymmetric Hypergroups In Proceedings of the 10th Internation Congress on Algebraic Hyperstruc-

tures and Applications Brno Czech Republic 3ndash9 September 2008 pp 303ndash309186 Massouros CG Mittas J On the theory of generalized M-polysymmetric hypergroups In Proceedings of the 10th Intern

Congress on Algebraic Hyperstructures and Applications Brno Czech Republic 3ndash9 September 2008 pp 217ndash228187 Massouros GG On the Hypergroup Theory Fuzzy Syst AI Rep Lett Acad Romana 1995 IV 13ndash25188 Sonea AC Cristea I The Class Equation and the Commutativity Degree for Complete Hypergroups Mathematics 2020 8

2253 [CrossRef]189 De Salvo M Lo Faro G On the n-complete hypergroups Discret Math 1999 208209 177ndash188 [CrossRef]190 De Salvo M Fasino D Freni D Lo Faro G 1-hypergroups of small size Mathematics 2021 9 108 [CrossRef]191 De Salvo M Freni D Lo Faro G A new family of hypergroups and hypergroups of type U on the right of size five Far East J

Math Sci 2007 26 393ndash418192 De Salvo M Freni D Lo Faro G On the hypergroups of type U on the right of size five with scalar identity J Mult Valued Log

Soft Comput 2011 17 425ndash441193 De Salvo M Fasino D Freni D Lo Faro G On strongly conjugable extensions of hypergroups of type U with scalar identity

Filomat 2013 27 977ndash994194 Fasino D Freni D Existence of proper semihypergroups of type U on the right Discret Math 2007 307 2826ndash2836 [CrossRef]195 Fasino D Freni D Minimal order semihypergroups of type U on the right Mediterr J Math 2008 5 295ndash314 [CrossRef]196 Freni D Structure des hypergroupes quotients et des hypergroupes de type U Ann Sci Univ Clermont-Ferrand II Math 1984

22 51ndash77197 Freni D Minimal order semihypergroups of type U on the right II J Algebra 2011 340 77ndash89 [CrossRef]198 Freni D Gutan M Sur les hypergroupes de type U Mathematica (Cluj) 1994 36 25ndash32199 Gutan M Sur une classe drsquohypergroupes de type C Ann Math Blaise Pascal 1994 1 1ndash19 [CrossRef]200 Gutan M Sureau Y Hypergroupes de type C lsquoa petites partitions Riv Mat Pura Appl 1995 16 13ndash38201 Heidari D Cristea I Breakable semihypergroups Symmetry 2019 11 100 [CrossRef]202 Heidari D Cristea I On Factorizable semihypergroups Mathematics 2020 8 1064 [CrossRef]203 Bordbar H Cristea I Regular Parameter Elements and Regular Local Hyperrings Mathematics 2021 9 243 [CrossRef]204 Massouros CG Separation and Relevant Properties in Hypergroups AIP Conf Proc 2016 1738 480051-1ndash480051-4 [CrossRef]205 Massouros CG On the semi-subhypergroups of a hypergroup Int J Math Math Sci 1991 14 293ndash304 [CrossRef]206 Massouros CG Some properties of certain subhypergroups Ratio Math 2013 25 67ndash76207 Freni D Ipergruppi ciclici e torsione negli ipergruppi Matematiche (Catania) 1980 35 270ndash286208 De Salvo M Freni D Sugli ipergruppi ciclici e completi Matematiche (Catania) 1980 35 211ndash226209 De Salvo M Freni D Semi-ipergruppi e ipergruppi ciclici Atti Sem Mat Fis Univ Modena 1981 30 44ndash59210 DeSalvo M Freni D LoFaro G Hypercyclic subhypergroups of finite fully simple semihypergroups J Mult Valued Log Soft

Comput 2017 29 595ndash617211 Vougiouklis T Cyclicity in a special class of hypergroups Acta Univ Carolin Math Phys 1981 22 3ndash6

40


212 Leoreanu V About the simplifiable cyclic semihypergroups Ital J Pure Appl Math 2000 7 69ndash76213 Al Tahan M Davvaz B On a special single-power cyclic hypergroup and its automorphisms Discret Math Algorithms Appl

2016 7 1650059 [CrossRef]214 Al Tahan M Davvaz B Commutative single power cyclic hypergroups of order three and period two Discret Math Algorithms

Appl 2017 9 1750070 [CrossRef]215 Al Tahan M Davvaz B On some properties of single power cyclic hypergroups and regular relations J Algebra Appl 2017 16

1750214 [CrossRef]216 Al Tahan M Davvaz B The cyclic hypergroup associated with Sn its automorphism group and its fuzzy grade Discret Math

Algorithms Appl 2018 10 1850070 [CrossRef]217 Al Tahan M Davvaz B Hypermatrix representations of single power cyclic hypergroups Ital J Pure Appl Math 2017

38 679ndash696218 Gu Z On cyclic hypergroups J Algebra Appl 2019 18 1950213 [CrossRef]219 Novak M Krehlik S Cristea I Cyclicity in EL-hypergroups Symmetry 2018 10 611 [CrossRef]220 Sureau Y Sous-hypergroupe engendre par deux sous-hypergroupes et sous-hypergroupe ultra-clos drsquoun hypergroupe C R

Acad Sc Paris 1977 284 983ndash984221 Haddad L Sureau Y Les cogroupes et les D-hypergroupes J Algebra 1988 118 468ndash476 [CrossRef]222 Haddad L Sureau Y Les cogroupes et la construction de Utumi Pac J Math 1990 145 17ndash58 [CrossRef]

41

mathematics

Article

Hypercompositional Algebra Computer Scienceand Geometry

Gerasimos Massouros 1 and Christos Massouros 2

1 School of Social Sciences Hellenic Open University Aristotelous 18 GR 26335 Patra Greece2 Core Department Euripus Campus National and Kapodistrian University of Athens GR 34400 Euboia

Greece ChrMasuoagr or ChMassourosgmailcom Correspondence germasourosgmailcom

Received 21 May 2020 Accepted 14 July 2020 Published 11 August 2020

Abstract The various branches of Mathematics are not separated between themselves On thecontrary they interact and extend into each otherrsquos sometimes seemingly different and unrelatedareas and help them advance In this sense the Hypercompositional Algebrarsquos path has crossedamong others with the paths of the theory of Formal Languages Automata and Geometry This paperpresents the course of development from the hypergroup as it was initially defined in 1934 by FMarty to the hypergroups which are endowed with more axioms and allow the proof of Theorems andPropositions that generalize Kleenrsquos Theorem determine the order and the grade of the states of anautomaton minimize it and describe its operation The same hypergroups lie underneath Geometryand they produce results which give as Corollaries well known named Theorems in Geometry likeHellyrsquos Theorem Kakutanirsquos Lemma Stonersquos Theorem Radonrsquos Theorem Caratheodoryrsquos Theoremand Steinitzrsquos Theorem This paper also highlights the close relationship between the hyperfields andthe hypermodules to geometries like projective geometries and spherical geometries

Keywords hypergroup hyperfield formal languages automata convex set vector space geometry

1 Introduction

This paper is written in the context of the special issue ldquoHypercompositional Algebra andApplicationsrdquo in ldquoMathematicsrdquo and it aims to shed light on two areas where the HypercompositionalAlgebra has expanded and has interacted with them Computer Science and Geometry

Hypercompositional Algebra is a branch of Abstract Algebra which appeared in the 1930s via theintroduction of the hypergroup

It is interesting that the group and the hypergroup are two algebraic structures which satisfyexactly the same axioms ie the associativity and the reproductivity but they differ in the law ofsynthesis In the first one the law of synthesis is a composition while in the second one it is ahypercomposition This difference makes the hypergroup a much more general algebraic structurethan the group and for this reason the hypergroup has been gradually enriched with further axiomswhich are either more powerful or less powerful leading thus to a significant number of specialhypergroups Among them there exist hypergroups that were proved to be very useful for the studyof Formal Languages and Automata as well as convexity in Euclidian vector spaces Furthermorebased on these hypergroups there derived other hypercompositional structures which are equally asuseful in the study of Geometries (spherical projective tropical etc) and Computer Science

A binary operation (or composition) ldquo middot rdquo on a non-void set E is a rule which assigns a unique elementof E to each element of Etimes E The notation ai middot a j = ak where ai a j ak are elements of E indicates thatak is the result of the operation ldquo middot rdquo performed on the operands ai and a j When no confusion arises

43


the operation symbol ldquo middot rdquo may be omitted and we write aia j = ak If the binary operation is associativethat is if it satisfies the equality

a(bc) = (ab)c for all a b c isin E

the pair (E middot) is called semigroup An element e isin E is an identity of (E middot) if for all a isin E ea = ae = aA triplet (E middot e) is a monoid if (E middot) is a semigroup and e is its identity If no ambiguity arises we candenote a semigroup (E middot) or a monoid (E middot e) simply by E A binary operation on E is reproductive ifit satisfies

aE = Ea = E for all a isin E

Definition 1 (Definition of the group) The pair (G middot) is called group if G is a non-void set and middot is an associativeand reproductive binary operation on G

This definition does not appear in group theory books but it is equivalent to the one mentionedin them We introduce it here as we consider it to be the most appropriate in order to demonstrate theclose relationship between the group and the hypergroup Thus the following two properties of thegroup derive from the above definition

Property 1 In a group G there is an element e called the identity element such that ae = ea = a for all a in G

Property 2 For each element a of a group G there exists an element aprime of G called the inverse of a such thata middot aprime = aprime middot a = e

The proof of the above properties can be found in [12] If no ambiguity arises we may abbreviate(G middot) to G

Example 1 (The free monoid) Often a finite non-empty set A is referred to as an alphabet The elements ofA1l ie the functions from 1 l to A are called strings (or words) of length l When l = 0 then wehave Aempty which is equal to empty The empty set empty is the only string of length 0 over A This string is called theempty string and it is denoted by λ If x is a string of length l over A and x(i) = ai i = 1 l we writex = (a1 a2 al) The set

strings(A) = Alowast =infin⋃

n=0

A1n

becomes a monoid if we define

(a1 a2 al)(b1 b2 bk) = (a1 a2 al b1 b2 bk)

The identity element of this binary operation which is called string concatenation is λ The strings of length1 generate the monoid since every element (a1 al) is a finite product (a1) (al) of strings of length 1The function g from A to A1 which is defined by

g(a) = (a) a isin A

is a bijection Thus we may identify a string (a) of length 1 over A with its only element a This means thatwe can regard the sets A1 and A as identical and consequently we may regard the elements of Alowast as wordsa1a2 al in the alphabet A It is obvious that for all x isin Alowast there is exactly one natural number n and exactlyone sequence of elements a1 a2 an of A such that x = a1a2 an Alowastis called the free monoid on the set (oralphabet) A

44


A generalization of the binary operation is the binary hyperoperation or hypercomposition ldquo middot rdquo on anon-void set E which is a rule that assigns to each element of Etimes E a unique element of the power setP(E) of E Therefore if ai a j are elements of E then ai middot a j sube E When there is no likelihood of confusionthe symbol ldquo middot rdquo can be omitted and we write aia j sube E If A B are subsets of E then A middot B signifies theunion

⋃(ab)isinAtimesB

a middot b In particular if A = empty or B = empty then AB = empty In both cases aA and Aa have the

same meaning as aA and Aa respectively Generally the singleton a is identified with its member a

Definition 2 (Definition of the hypergroup) The pair (H middot) is called hypergroup if H is a non-void set and middot isan associative and reproductive binary hypercomposition on H

The following Proposition derives from the definition of the hypergroup

Proposition 1 In a hypergroup the result of the hypercomposition of any two elements is non-void

The proof of Proposition 1 can be found in [34] If no ambiguity arises we may abbreviate (H middot) toH Two significant types of hypercompositions are the closed and the open ones A hypercompositionis called closed [5] (or containing [6] or extensive [7]) if the two participating elements always belong tothe result of the hypercomposition while it is called open if the result of the hypercomposition of anytwo elements different from each other does not contain the two participating elements

The notion of the hypergroup was introduced in 1934 by F Marty who used it in order to studyproblems in non-commutative algebra such as cosets determined by non-invariant subgroups [8ndash10]From Definitions 1 and 2 it is evident that both groups and hypergroups satisfy the same axiomsand their only difference is that the law of synthesis of two elements is a composition in groupswhile it is a hypercomposition in hypergroups This difference makes the hypergroups much moregeneral algebraic structures than the groups to the extent that properties similar to the previous 1and 2 generally cannot be proved for the hypergroups Furthermore in the hypergroups there existdifferent types of identities [11ndash13] In general an element e isin H is an identity if a isin aecap ea for all a isin HAn identity is called scalar if a = ae = ea for all a isin H while it is called strong if a isin ae = ea sube e a forall a isin H Obviously if the hypergroup has an identity then the hypercomposition cannot be open

Besides in groups both equations a = xb and a = bx have a unique solution while in thehypergroups the analogous relations a isin xb and a isin bx do not have unique solutions Thus F Martyin [8] defined the two induced hypercompositions (right and left division) that derive from thehypergrouprsquos hypercomposition

a| b = x isin H | a isin xb and

ab | = x isin H | a isin bx

If H is a group then a| b = abminus1 and a

b | = bminus1a It is obvious that if ldquordquo is commutative thenthe right and the left division coincide For the sake of notational simplicity a b or a b is used todenote the right division or right hyperfraction or just the division in the commutative hypergroupsand b a or ab is used to denote the left division or left hyperfraction [1415] Using the inducedhypercomposition we can create an axiom equivalent to the reproductive axiom regarding whichthe following Proposition is valid [34]

Proposition 2 In a hypergroup H the non-empty result of the induced hypercompositions is equivalent to thereproductive axiom

45


W Prenowitz enriched a commutative hypergroup of idempotent elements with one more axiomin order to use it in the study of Geometry [16ndash21] More precisely in a commutative hypergroup Hall the elements of which satisfy the properties aa = a and a a = a he introduced the transposition axiom

a bcap c d empty implies adcap bc empty for all a b c d isin H

He named this new hypergroup join space For the sake of terminology unification a commutativehypergroup which satisfies the transposition axiom is called join hypergroup Prenowitz was followed byJ Jantosciak [20ndash23] V W Bryant R J Webster [24] D Freni [2526] J Mittas C G Massouros [227ndash29]A Dramalidis [3031] etc In the course of his research J Jantosciak generalized the transpositionaxiom in an arbitrary hypergroup as follows

b acap c d empty implies adcap bc empty for all a b c d isin H

and he named this hypergroup transposition hypergroup [15] These algebraic structures attracted theinterest of a big number of researchers among whom there are J Jantosciak [152223] I Cristea [32ndash35]P Corsini [35ndash42] V Leoreanu-Fortea [40ndash46] S Hoskova-Mayerova [47ndash51] J Chvalina [48ndash52]P Rackova [4950] C G Massouros [36121353ndash62] G G Massouros [3121353ndash62] R Ameri [63ndash65]M M Zahedi [63] I Rosenberg [66] etc

Furthermore it has been proved that these hypergroups also comprise a useful tool in the studyof Languages and Automata [67ndash71] and a constructive origin for the development of other newhypercompositional structures [5357587273]

Definition 3 A transposition hypergroup which has a scalar identity e is called quasicanonicalhypergroup [7475] or polygroup [76ndash78]

In the quasicanonical hypergroups there exist properties analogous to 1 and 2 which are valid inthe groups

Proposition 3 [1553] If (Q middot e) is a quasicanonical hypergroup then

(i) for each a isin Q there exists one and only one aprime isin Q such that e isin aaprime = aprimea(ii) c isin abrArr a isin cbprime rArr b isin aprimec

The inverse is also true

Proposition 4 [1553] If a hypergroup Q has a scalar identity e and



A commutative quasicanonical hypergroup is called canonical hypergroup This hypergroup wasfirst used by M Krasner [79] but it owes its name to J Mittas [8081]

A non-empty subset K of a hypergroup H is called semi-subhypergroup when it is stable underthe hypercomposition ie xy sube K for all x y isin K K is a subhypergroup of H if it satisfies thereproductive axiom ie if the equality xK = Kx = K is valid for all x isin K Since the structureof the hypergroup is much more complicated than that of the group there are various kinds ofsubhypergroups A subhypergroup K of H is called closed from the right (in H) (resp from the left) iffor every element x in the complement Kc of K it holds that (xK) capK = empty (resp(Kx) capK = empty) K is

46


called closed if it is closed both from the right and from the left [82ndash84] It has been proved [1] that asubhypergroup is closed if and only if it is stable under the induced hypercompositions ie

a b sube K and b a sube K for all a b isin K

A subhypergroup K of a hypergroup is invertible if a b cap K empty implies b acapK emptyand b acapK empty implies a b cap K empty From this definition it derives that every invertiblesubhypergroup is also closed but the opposite is not valid

Proposition 5 [114] If a subset K of a hypergroup H is stable under the induced hypercompositions then K is asubhypergroup of H

Proposition 6 [114] If K is a closed subhypergroup of a hypergroup H and a isin K then

a K = K a = aK = K = Ka = K a = a K

It has been proved [146062] that the set of the semi-subhypergroups (resp the set of the closedsubhypergroups) which contains a non-void subset E is a complete lattice Hence the minimum (inthe sense of inclusion) semi-subhypergroup of a hypergroup H which contains a given non-emptysubset E of H can be assigned to E This semi-subhypergroup is denoted by [E] and it is called thegenerated by E semi-subhypergroup of H Similarly 〈E〉 is the generated by E closed subhypergroupof H For notational simplicity if E = a1 an then [E] = [a1 an] and 〈E〉 = 〈a1 an〉 areused instead

Duality Two statements of the theory of hypergroups are dual statements (see [1553]) if each one ofthem results from the other by interchanging the order of the hypercomposition ldquomiddotrdquo that is interchangingany hyperproduct ab with ba Observe that the reproductive and the associative axioms are self-dualMoreover observe that the induced hypercompositions and have dual definitions hence they mustbe interchanged during the construction of a dual statement So the transposition axiom is self-dual aswell Therefore the following principle of duality holds for the theory of hypergroups

Given a Theorem the dual statement which results from the interchange of the order of thehypercomposition (and the necessary interchange of and ) is also a Theorem

Special notation In the following pages apart from the typical algebraic notations we are usingKrasnerrsquos notation for the complement and difference So we denote with AB the set of elements thatare in the set A but not in the set B

2 Formal Languages Automata Theory and Hypercompositional Structures

Mathematically a language whose words are written with letters from an alphabet Σ is definedas a subset of the free monoid Σ generated by Σ The above definition of the language is fairly generaland it includes all the written natural languages as well as the artificial ones In general a language isdefined in two ways It is either presented as an exhaustive list of all its valid words ie through adictionary or it is presented as a set of rules defining the acceptable words Obviously the first methodcan only be used when the language is finite All the natural languages such as English or Greek arefinite and they have their own dictionaries Artificial languages on the other hand may be infiniteand they can only be defined by the second way

In the artificial languages precision and no guesswork are required especially when computers areconcerned The regular expressions which are very precise language-defining symbols were createdand developed as a language-defining symbolism The languages that are associated with theseexpressions are called regular languages

47


The regular expressions were introduced by Kleene [85] who also proved that they are equivalentin expressive power to finite automata McNaughton and Yamada gave their own proof to this [86]while Brzozowski [8788] and Brzozowski and McClusky [89] further developed the theory of regularexpressions In regular languages the expression x + y where x and y are strings of characters from analphabet Σ means ldquoeither x or yrdquo Therefore x + y =

x y

In this way the monoid Σ is enriched with

a hypercomposition This hypercomposition is named B-hypercomposition [67ndash69]

Proposition 7 [6768] A non-void set equipped with the B-hypecomposition is a join hypergroup

A hypergroup equipped with the B-hypercomposition is called B-hypergroup [67ndash69] Moreoverthe empty set of words and its properties in the theory of the regular languages leads to the followingextension Let 0 lt Σ In the set Σ = Σcup0 a hypercomposition called dilated B-hypercompositionis defined as follows

x + y =x y

if x y isin Σ and x y

x + x = x 0 for all x isin Σ

The associativity and the commutativity of the dilated B-hypercomposition derive withoutdifficulty Moreover the transposition axiom is verified since

x y = y x =

H i f x = yx y

i f x y and x = e

x i f x y and x e

This join hypergoup is called dilated B-hypergroup and it has led to the definition of a new class ofhypergroups the class of the fortified transposition hypergroups and fortified join hypergroups

An automaton A is a collection of five objects (Σ S δ so F) where Σ is the alphabet of inputletters (a finite nonempty set of symbols) S is a finite nonvoid set of states so is an element of Sindicating the start (or initial) state F is a (possibly empty) subset of S representing the set of thefinal (or accepting) states and δ is the state transition function with domain Stimes Σ and range S in thecase of a deterministic automaton (DFA) or P(S) the powerset of S in the case of a nondeterministicautomaton (NDFA) Σ denotes the set of words (or strings) formed by the letters of Σ ndashclosure of Σndashand λ isin Σ signifies the empty word Given a DFAA the extended state transition function forAdenoted δ is a function with domain Stimes Σ and range S defined recursively as follows

i δ(s a) = δ(s a) for all s in S and a in Σii δ(sλ) = s for all s in Siii δ(s ax) = δ(δ(s a) x) for all s in S x in Σ and a in Σ

In [67ndash71] it is shown that the set of the states of an automaton equipped with differenthypercompositions can be endowed with the structure of the hypergroup The hypergroups that havederived in this way are named attached hypergroups to the automaton To date various types of attachedhypergroups have been developed to represent the structure and operation of the automata with theuse of the hypercompositional algebra tools Between them are

i the attached hypergroups of the order andii the attached hypergroups of the grade

Those two types of hypergroups were also used for the minimisation of the automata In additionin [69] another hypergroup derived from a different definition of the hypercomposition was attachedto the set of the states of the automaton Due to its definition this hypergroup was named the attachedhypergroup of the paths and it led to a new proof of Kleenersquo s Theorem Furthermore in [70] the attachedhypergroup of the operation was defined in automata One of its applications is that this hypergroup canindicate all the states on which an automaton can be found after the t-clock pulse For the purpose of

48


defining the attached hypergroup of the operation the notions of the Prefix and the Suffix of a wordneeded to be introduced Let x be a word in Σ then

Pre f ix(x) =y isin Σ

∣∣∣ yz = x for some z isin Σ

and Su f f ix(x) =z isin Σ

∣∣∣ yz = x for some y isin Σ

Let s be an element of S Then

Is =x isin Σ

∣∣∣ δ(so x) = s

and Ps =

si isin S∣∣∣ si = δ(so y) y isin Pre f ix(x) x isin Is

Obviously the states so and s are in Ps

Lemma 1 If r isin Ps then Pr sube Ps

Proof Pr =si isin S

∣∣∣ si = δ(so y) y isin Pre f ix(v) v isin Ir

and since r isin Ps it holds that δ(r z) = sfor some z in Su f f ix(x) x isin Is Thus δ(si yi) = s yi isin Su f f ix and so the Lemma

With the use of the above notions more hypercompositional structures can be attached on the setof the states of the automaton

Proposition 8 The set S of the states of an automaton equipped with the hypercomposition

s + q = Ps

⋃Pq f or all s q isin S

becomes a join hypergroup

Proof Initially notice that s + S =⋃qisinS

(Ps cup Pq) = S Hence the reproductive axiom is valid Next

the definition of the hypercomposition yields the equality

s + (q + r) = s + (Pq cup Pr) = Ps cup (⋃

uisinPqcupPr

Pu)

Per Lemma 1 the right-hand side of the above equality is equal to Ps cup (Pq cup Pr) which howeveris equal to (Ps cup Pq)cup Pr Using again Lemma 1 we get the equality (Ps cup Pq)cup Pr = (

⋃visinPscupPq

Pv)cup Pr

Thus(

⋃

visinPscupPq

Pv)cup Pr = (Ps cup Pq) + r = (s + q) + r

and so the associativity is valid Next observe that the hypercomposition is commutative and therefore

s q = q s =

S i f s isin Pq

r isin S | Ps sube Pr i f s lt Pq

Suppose that s qcap p r empty Then (s + r)cap (q + p) = (Ps cup Pr)cap (Pq cup Pp) which is non-emptysince it contains so Hence the transposition axiom is valid and so the Proposition


s + q = Ps cap Pq f or all s q isin S

becomes a join semihypergroup

49


Proof Since so isin Pr for all r isin S the result of the hypercomposition is always non-void On theother hand

s q = q s =

S i f s isin Pq

empty i f s lt Pq

hence since s q with s q in S is not always nonvoid the reproductive axiom is not validThe associativity can be verified in the same way as in the previous Proposition Finally if s qcapp r emptythen s qcap p r = S and so the intersection (s + r)cap (q + p) which is equal to (Ps cap Pr)cap (Pq cap Pp) isnon-empty since it contains so

G G Massouros [68ndash73] G G Massouros and J D Mittas [67] and after them J Chvalina [90ndash92]L Chvalinova [90] M Novak [91ndash94] S Krehliacutek [91ndash93] M M Zahedi [95] M Ghorani [9596] etcstudied automata using algebraic hypercompositional structures

Formal Languages and Automata theory are very close to Graph theory P Corsini [9798] MGionfriddo [99] Nieminen [100101] I Rosenberg [66] M De Salvo and G Lo Faro [102ndash104] I Cristeaet al [105ndash108] C Massouros and G Massouros [109110] C Massouros and C Tsitouras [111112]and others studied hypergroups associated with graphs In the following we will present how toattach a join hypergroup to a graph In general a graph is a set of points called vertices connected bylines which are called edges A path in a graph is a sequence of no repeated vertices v1 v2 vnsuch that v1v2 v2v3 vnminus1vn are edges in the graph A graph is said to be connected if every pair ofits vertices is connected by a path A tree is a connected graph with no cycles Let T be a tree In the setV of its vertices a hypercompostion ldquomiddotldquo can be introduced as follows for each two vertices x y in Vx middot y is the set of all vertices which belong to the path that connects vertex x with vertex y Obviouslythis hypercomposition is a closed hypercomposition ie x y are contained in x middot y for every x y in V

Proposition 10 If V is the set of the vertices of a tree T then (V middot) is a join hypergroup

Proof Sincex y

sube xy it derives that xV = V for each x in V and therefore the reproductive axiomis valid Moreover since T is an undirected graph the hypecomposition is commutative Nextlet x y z be three arbitrary vertices of T If any of these three vertices eg z belongs to the paththat the other two define then (xy)z = x(yz) = xy If x y z do not belong to the same path thenthere exists only one vertex v in xy such that vzcap xy = v Indeed if there existed a second vertexw such that wzcap xy = w then the tree T would have a cycle which is absurd So (xy)z = xycup vzand x(yz) = xvcup yz Since xycup vz = xvcup yz it derives that (xy)z = x(yz) Now for the transpositionaxiom suppose that x y z w are vertices of T such that x ycap z w empty If x y z w are in thesame path then considering all their possible arrangements in their path it derives that xwcap yz emptyNext suppose that the four vertices do not belong to the same path Thus suppose that z does notbelong to the path defined by y w Then z lt yw Consider zy and zw As indicated above since thereare no cycles in T there exists only one vertex v in xy such that zy = yvcup vz and zw = wvcup vz Nowwe distinguish between the cases

(i) if x y w do not belong to the same path then for the same reasons as above there exists only ones in xy such that xy = yscup sx and sw = wscup sx Since x ycap z w empty there exists r in V suchthat x isin ry and z isin rw Thus since T contains no cycles and in order for srv not to form a cycle sand v must coincide Hence v isin xwcap yz and therefore xwcap yz empty

(ii) if x belongs to the same path with y and w then

(iia) if x isin yw then yw = yx cup xw and xw sube x y Hence v = x x isin xw cap yz and thereforexwcap yz empty

(iib) if x lt yw then x ycap z w = empty

50


A spanning tree of a connected graph is a tree whose vertex set is the same as the vertex set ofthe graph and whose edge set is a subset of the edge set of the graph Any connected graph has atleast one spanning tree and there exist algorithms which find such trees Hence any connected graphcan be endowed with the join hypergroup structure through its spanning trees Moreover since aconnected graph may have more than one spanning trees more than one join hypergroups can beassociated to a graph On the other hand in any connected or not connected graph a hypergroup canbe attached according to the following Proposition

Proposition 11 The set V of the vertices of a graph is equipped with the structure of the hypergroup if theresult of the hypercomposition of two vertices vi and v j is the set of the vertices which appear in all the possiblepaths that connect vi to v j or the set

vi v j

if there do not exist any connecting paths from vertex vi to

vertex v j

21 Fortified Transposition Hypergroups

Definition 4 A fortified transposition hypergroup (FTH) is a transposition hypergroup H with a unique strongidentity e which satisfies the axiom

for every x isin He there exists one and only one element y isin He such that e isin xy and e isin yxy is denoted by xminus1 and it is called inverse or symmetric of x When the hypercomposition is written additivelythe strong identity is denoted by 0 the unique element y is called opposite or negative instead of inverseand the notation minusx is used If the hypercomposition is commutative the hypergroup is called fortified joinhypergroup (FJH)

It has been proved that every FTH consists of two types of elements the canonical (c-elements) andthe attractive (a-elements) [5357] An element x is called canonical if ex = xe is the singleton x while itis called attractive if ex = xe = e x We denote with A the set of the a-elements and with C the set ofthe c-elements By convention e isin A

Proposition 12 [5357]

(i) if x is a non-identity attractive element then e x = e middot xminus1 =xminus1 e

= xminus1 middot e = x e

(ii) if x is a canonical element then e x = xminus1 = x e

(iii) if x y are attractive elements and x y then x middot yminus1 = x ycupyminus1

and yminus1 middot x = y xcup

yminus1

(iv) if x y are canonical elements then x middot yminus1 = x y and yminus1 middot x = y x

Theorem 1 [57]

(i) the result of the hypercomposition of two a-elements is a subset of A and it always contains thesetwo elements

(ii) the result of the hypercomposition of two non-symmetric c-elements consists of c-elements(iii) the result of the hypercomposition of two symmetric c-elements contains all the a-elements(iv) the result of the hypercomposition of an a-element with a c-element is the c-element

Theorem 2 [5357] If H is a FTH then the set A of the attractive elements is the minimum (in the sense ofinclusion) closed subhypergroup of H

The proof of the above Theorems as well as other properties of the theory of the FTHs and FJHscan be found in [5355ndash5761] The next two Propositions refer to the reversibility in FTHs

Lemma 2 If w isin x middot y then w middot xminus1 cap e middot y empty and w middot yminus1 cap e middot x empty

51


Proof w isin x middot y implies x isin w y and y isin x w Moreover x isin xminus1 e and y isin e y minus1 Consequentlyw ycap xminus1 e empty and x wcap y isin e yminus1 empty Next the transposition axiom gives the Lemma

Proposition 13 If w isin x middot y and if any one of x y is a canonical element then

x isin w middot yminus1 and y isin xminus1 middotw

Proof We distinguish between two cases

(i) If x y isin C then e middot x = x and e middot y = y Next Lemma 2 applies and yields the Proposition(ii) Suppose that x isin A and y isin C Then according to Theorem 1(iv) x middot y = y thus w = y Via

Theorem 1(iii) A sube y middot yminus1 thus x isin y middot yminus1 Per Theorem 2 xminus1 isin A consequently y = xminus1y

Hence the Proposition is proved

Proposition 14 Suppose that x y are attractive elements and w isin x middot y

(i) if w = y = e then e isin xe implies that e isin ex while x lt ee(ii) if w = x y then x isin x middot y implies that x isin xyminus1 while generally y lt xxminus1

(iii) in any other case w isin x middot y implies x isin w middot yminus1 and y isin xminus1 middotw

Sketch of Proof Cases (i) and (ii) are direct consequences of the Theorem 1 while case (iii) derivesfrom the application of Lemma 2

The property which is described in Proposition 13 is called reversibility and because of Proposition 14this property holds partially in the case of TPH

Another distinction between the elements of the FTH stems from the fact that the equality(xxminus1

)minus1= xxminus1 (or minus(xminus x) = xminus x in the additive case) is not always valid The elements that satisfy

the above equality are called normal while the others are called abnormal [5357]

Example 2 Let H be a totally ordered set dense and symmetric around a center denoted by 0 isin H With regardsto this center the partition H = Hminus cup 0 cupH+ can be defined according to which for every x isin Hminus andy isin H+ it is x lt 0 lt y and x le yrArr minusy le minusx for every x y isin H where ndashx is the symmetric of x with regardsto 0 Then H equipped with the hypercomposition

x + y =x y

i f y minusx

andx + (minusx) = [ 0 |x| ] cup minus|x|

becomes a FJH in which xminus x minus(xminus x) for every x 0

Proposition 15 The canonical elements of a FTH are normal

Proof Let x be a canonical element Because of Theorem 1 A sube xxminus1 while according to Theorem 2

Aminus1 = A Thus Aminus1 sube xxminus1 and therefore A sube(xxminus1

)minus1 Suppose that z is a canonical element in xxminus1

Per Proposition 13 x isin zx So xxminus1 sube z(xxminus1

) Hence we have the sequence of implications

e isin z(xxminus1

) zminus1 isin xxminus1 z isin

(xxminus1

)minus1

52


So xxminus1 sube(xxminus1

)minus1 Furthermore xxminus1 sube

(xxminus1

)minus1implies that

(xxminus1

)minus1 sube[(

xxminus1)minus1

]minus1= xxminus1 and

therefore the Proposition holds

An important Theorem that is valid for TFH [53] is the following structure Theorem

Theorem 3 A transposition hypergroup H containing a strong identity e is isomorphic to the expansion ofthe quasicanonical hypergroup Ccup e by the transposition hypergroup A of all attractive elements through theidentity e

The special properties of the FTH give different types of subhypergroups There existsubhypergroups of a FTH that do not contain the symmetric of each one of their elements while thereexist others that do This leads to the definition of the symmetric subhypergroups A subhypergroup Kof a FTH is symmetric if x isin K implies xminus1 isin K It is known that the intersection of two subhypergroupsis not always a subhypergroup In the case of the symmetric subhypergroups though the intersectionof two such subhypergroups is always a symmetric subhypergoup [5762] Therefore the set ofthe symmetric subhypergroups of a FTH consist a complete lattice It is proved that the lattice ofthe closed subhypergroups of a FTH is a sublattice of the lattice of the symmetric subhypergroupsof the FTH [5762] An analytic and detailed study of these subhypergroups is provided in thepapers [16062] Here we will present the study of the monogene symmetric subhypergroups iesymmetric subhypergroups generated by a single element So let H be a FJH let x be an arbitraryelement of H and let M(x) be the monogene symmetric subhypergroup which is generated by thiselement Then

xn =

x middot x middot x (n times) i f n gt 0e i f n = 0xminus1 middot xminus1 middot xminus1 (minusn times) i f n lt 0

(1)

and

xm middot xn =

xm+n i f mn gt 0

xm+n middot(x middot xminus1

)min|m||n|i f mn lt 0

(2)

From the above it derives thatxm+n sube xm middot xn

Theorem 4 If x is an arbitrary element of a FJH then the monogene symmetric subhypergroup which isgenerated by this element is

M(x) =⋃

(mn)isinZtimesN0

xm middot(x middot xminus1

)n

Proof The symmetric subhypergroup of a normal FTH which is generated from a non-empty setX consists of the unions of all the finite products of the elements that are contained in the unionXminus1 cupX [62] thus from (1) we have

M(x) =⋃

(kl)isinN2

xk middot(xminus1

)l=

⋃

(kl)isinN2

xk middot xminusl

According to (2) it is xk middot xminusl = xkminusl middot(x middot xminus1

)mink l But k minus l isin Z therefore the Theorem

is established

From the above Theorem Proposition 13 and Theorem 1 we have the following Corollary

53


Corollary 1 Every monogene symmetric subhypergroup M(x) with generator a canonical element x is closed itcontains all the attractive elements and also

M(x) =⋃

(mn)isinZ2


)n

Remark 1

(i) Since e isin x middot xminus1 the inclusion xm middot(x middot xminus1

)n sube xm middot(x middot xminus1

)qis valid for n lt q

(ii) for x = e it is M(e) = e

Let us define now a symbol ω(x) (which can even be the +infin) and name it order of x andsimultaneously order of the monogene subhypergroup M(x) Two cases can appear such that one revokesthe other

I For any (m n) isin ZtimesN0 with m 0 we have

e lt xm middot(x middot xminus1

)n

Then we define the order of x and of M(x) to be the infinity and we write ω(x) = +infin

Proposition 16 If ω(x) = +infin then x is a canonical element

Proof Suppose that x belongs to the set A of the attractive elements Then per Theorem 1(i) xm sube Aand x isin xm Consequently

e isin x middot e sube xm middot e sube xm middot(x middot xminus1

)n

This contradicts our assumption and therefore x is a canonical element

The previous Proposition and Theorem 1 result to the following Corollary

Corollary 2 If ω(x) = +infin then xm does not contain attractive elements for every m isin Z

Proposition 17 If ω(x) = +infin then

xm+n cap xn = empty i f m gt 0 and xnminusm cap xn = empty i f m lt 0

for any (m n) isin ZtimesN0 with m 0

Proof From e lt xm middot(x middot xminus1

)nit derives that xm cap

(x middot xminus1

)minusn= empty According to Proposition 16 x is a

canonical element and therefore because of Proposition 15 x is normal thus(x middot xminus1

)minusn=

(x middot xminus1

)n

Therefore xm cap(x middot xminus1

)n= empty or xm cap (xn middot xminusn) = empty So the Proposition follows from the reversibility

Proposition 18 ω(x) = +infin if and only if

(i) xm sube C for every m isin Z(ii) xm1 cap xm2 = empty for every m1 m2 isin Z with m1 m2

54


Proof If ω(x) = +infin per Corollary 2 xm does not contain attractive elements for every m isin ZMoreover if n 0 then (ii) derives from Proposition 17 If n = 0 then e lt xm and assuming thatm = m1 + m2 with m1m2 gt 0 we successively have

e lt xm e lt xm1+m2 e lt xm1xm2 xminusm1 cap xm2 = empty

Conversely now If for every m isin Z the intersection xm capA is void and if for every m1 m2 isin Zwith m1 m2 the intersection xm1 cap xm2 is also void then e lt xm1xminusm2 and therefore

e lt xm1minusm2 i f m1m2 lt 0

ande lt xm1minusm2

(xxminus1

)min|m1 | |m2 | i f m1m2 gt 0

Thuse lt xm middot

(x middot xminus1

)nfor every (m n) isin ZtimesN0

So the Proposition holds

II There exist (m n) isin ZtimesN0 with m 0 such that

e isin xm middot(x middot xminus1

)n

Proposition 19 Let p be the minimum positive integer for which there exists s isin N0 such that e isin xp middot(x middot xminus1

)s

Then for a given m isin Z there exist n isin N such that e isin xm middot(x middot xminus1

)nif and only if m is divided by p

Proof Let m = kp k isin Z From e isin xp middot(x middot xminus1

)rit derives that

e isin xkp middot(x middot xminus1

)kr= xm middot

(x middot xminus1

)n

Therefore the PropositionConversely now If x is an a-element then e isin x middot

(x middot xminus1

)nfor every n isin N so p = 1 and thus the

Proposition Next if x is a c-element and e isin xm middot(x middot xminus1

)nwith m = kp + r k isin Z 0 lt r lt p Then


)n= xkp+r middot

(x middot xminus1

)n sube xkp middot xr middot(x middot xminus1

)n

According to our hypothesis e isin xp middot(x middot xminus1

)s Moreover per Theorem 1 the sum of two

non-opposite c-elements does not contain any a-elements Consequently there do not exist a-elementsin xp and so

xminusp sube xminusp middot xp middot(x middot xminus1

)n=

(xminus1 middot x

)p middot(x middot xminus1

)n=

(x middot xminus1

)p+n

Thusxkp sube

(x middot xminus1

)minusk(p+n)=

(x middot xminus1

)| k|(p+n)

and thereforee isin

(x middot xminus1

)| k|(p+n) middot xr middot(x middot xminus1

)n= xr middot

(x middot xminus1

)| k|(p+n)+n

This contradicts the supposition according to which p is the minimum element with the propertye isin xp middot

(x middot xminus1

)s Thus r = 0 and so m = kp

55


For m = kp k isin Z let qk be the minimum non-negative integer for which e isin xkp middot(x middot xminus1

)qk Thusa function q Zrarr N0 is defined which corresponds each k in Z to the non-negative integer qk

Definition 5 The pair ω(x) = (p q) is called order of x and of M(x) The number p is called principal orderof x and of M(x) while the function q is called associated order of x and of M(x)

Consequently according to the above definition if x is an attractive element then e isin x middot(x middot xminus1

)

and therefore ω(x) = (1 q) with q(k) = 1 for every k isin Z Moreover if x is a self-inverse canonical

element then e isin x2 middot(x middot xminus1

)0 if x lt x middot xminus1 and e isin x middot

(x middot xminus1

) if x isin x middot xminus1 and thus ω(x) = (2 q) with

q(k) = 0 in the first case and ω(x) = (1 q) with q(k) = 1 in the second case (for every k isin Z)Moreover we remark that the order of e is ω(e) = (1 q) with q(k) = 0 for every k isin Z and e is

the only element which has this property Yet it is possible that there exist non-identity elements x isin Hwith prime order 1 and this happens if and only if there exists an integer n such that xminus1 isin

(x middot xminus1

)n

as for example when x is a self-inverse canonical element

22 The Hyperringoid

Let Σ be the set of strings over an alphabet Σ Then

Proposition 20 String concatenation is distributive over the B-hypecomposition

Proof Let a b c isin Σlowast Then a(b + c) = ab c = ab ac = ab + ac

Via the thorough verification of the distributive axiom in all the different cases and taking intoconsideration that 0 is a bilaterally absorbing element with respect to the string concatenation on theset Σ it can also be proved that

Proposition 21 String concatenation is distributive over the dilated B-hypecomposition

Consequently Σlowast and Σ are algebraic structures equipped with a composition and ahypercomposition which are related with the distributive law

Definition 6 A hyperringoid is a non-empty set Y equipped with an operation ldquomiddotrdquo and a hyperoperation ldquo+rdquosuch that

i (Y+) is a hypergroupii (Y middot) is a semigroupiii the operation ldquomiddotrdquo distributes on both sides over the hyperoperation ldquo+ rdquo

If the hypergroup (Y+) has extra properties which make it a special hypergroup it gives birth tocorresponding special hyperringoids So if (Y+) is a join hypergroup then the hyperringoid is calledjoin A distinct join hyperringoid is the B-hyperringoid in which the hypergroup is a B-hypergroupA fortified join hyperringoid or join hyperring is a hyperringoid whose additive part is a fortified joinhypergroup and whose zero element is bilaterally absorbing with respect to the multiplication A specialjoin hyperring is the join B-hyperring in which the hypergroup is a dilated B-hypergroup If theadditive part of a fortified join hyperringoid becomes a canonical hypergroup then it is called hyperringThe hyperrigoid was introduced in 1990 [67] as the trigger for the study of languages and automata withthe use of tools from hypercompositional algebra An extensive study of the fundamental properties ofhyperringoids can be found in [61113ndash116]

Example 3 Let (R+ middot) be a ring If in R we define the hypercomposition

aoplus b = a b a + b f or all a b isin R

56


then (Roplus middot) is a join hyperring

Example 4 Let le be a linear order (also called a total order or chain) on Y ie a binary reflexive and transitiverelation such that for all y yprime isin Y y yprime either y le yprime or yprime le y is valid For y yprime isin Y y lt yprime the setz isin Y

∣∣∣ y le z le yprime

is denoted by [y yprime] and the setz isin Y

∣∣∣ y lt z lt yprime

is denoted by ]y yprime[ The order isdense if no ]y yprime[ is void Suppose that (Y middotle) is a totally ordered group ie (Y middot) is a group such that forall y le yprime and x isin Y it holds that x middot y le x middot yprime and y middot x le yprime middot x If the order is dense then the set Y can beequipped with the hypercomposition

x + y =

x i f x = y]min

x y

max

x y

[ i f x y

and the triplet (Y+ middot) becomes a join hyperringoid Indeed since the equalities

x + y = ]minx y

max

x y

[ = y + x

and(x + y) + z = ]min

x y z

max

x y z

[ = x + (y + z)

are valid for every x y z isin Y the hypercomposition is commutative and associative Moreover

x y = y x =

x i f x = yt isin Y x lt t i f y lt xt isin Y t lt x i f x lt y

Thus when the intersection (x y)cap (z w) is non-void the intersection (x + w)cap (z + y) is also non-voidSo the transposition axion is valid Therefore (Y+) is a join hypergroup Moreover

x middot (y + z) =

x middot y = x middot y + x middot z i f y = zx middot ]y z[ = x middot ⋃

ylttltzt = ⋃

ylttltzx middot t = x middot y + x middot z i f y z

It is worth mentioning that the hypercomposition

x + y = [minx y

max

x y

] f or all x y isin Y

endows (Y middot) with the join hyperringoid structure as well

As per Proposition 20 the set of the words Σ over an alphabet Σ can be equipped with thestructure of the B-hyperringoid This hyperringoid has the property that each one of its elementswhich are the words of the language has a unique factorization into irreducible elements which arethe letters of the alphabet So this hyperringoid has a finite prime subset that is a finite set of initialand irreducible elements such that each one of its elements has a unique factorization with factorsfrom this set In this sense this hyperringoid has a property similar to the one of the Gaussrsquo ringsMoreover because of Proposition 21 Σ can be equipped with the structure of the join B-hyperringwhich has the same property

Definition 7 A linguistic hyperringoid (resp linguistic join hyperring) is a unitary B-hyperringoid (resp joinB-hyperring) which has a finite prime subset P and which is non-commutative for |P| gt 1

It is obvious that every B-hyperringoid or join B-hyperring is not a linguistic one

Proposition 22 From every non-commutative free monoid with finite base there derives a linguistic hyperringoid

57


Example 5 Let 0 12times2 express the set of 2 times 2 matrices which consist of the elements 01 that is the following16 matrices

A1 =

[1 00 0

] A2 =

[0 10 0

] A3 =

[0 01 0

] A4 =

[0 00 1

]

B1 =

[1 10 0

] B2 =

[0 10 1

] B3 =

[0 01 1

] B4 =

[1 01 0

] B5 =

[1 00 1

] B6 =

[0 11 0

]

C1 =

[0 11 1

] C2 =

[1 01 1

] C3 =

[1 11 0

] C4 =

[1 10 1

]

D1 =

[1 11 1

]

E1 =

[0 00 0

]

Consider the set T of all 2 times 2 matrices deriving from products of the above matrices except the zero matrixT becomes a B-hyperringoid under B-hypercomposition and matrix multiplication Observe that none of thematrices A1 B6 C4 can be written as the product of any two matrices from the set T while all the matrices in Tresult from products of these three matrices Therefore T is a linguistic hyperringoid whose prime subset isA1 B6 C4 Furthermore if T is enriched with the zero matrix then it becomes a linguistic join hyperring

M Krasner was the first one who introduced and studied hypercompositional structureswith an operation and a hyperoperation The first structure of this kind was the hyperfieldan additive-multiplicative hypercompositional structure whose additive part is a canonical hypergroupand the multiplicative part a commutative group The hyperfield was introduced by M Krasnerin [79] as the proper algebraic tool in order to define a certain approximation of complete valuedfields by sequences of such fields Later on Krasner introduced the hyperring which is related to thehyperfield in the same way as the ring is related to the field [117] Afterwards J Mittas introduced thesuperring and the superfield in which both the addition and the multiplication are hypercompositionsand more precisely the additive part is a canonical hypergroup and the multiplicative part is asemi-hypergroup [118ndash120] In the recent bibliography a structure whose additive part is a hypergroupand the multiplicative part is a semi-group is also referred to with the term additive hyperring andsimilarly the term multiplicative hyperring is used when the multiplicative part is a hypergroup

Rings and Krasnerrsquos hyperrings have many common elementary algebraic properties eg in bothstructures the following are true

(i) x(minusy) = (minusx)y = minusxy(ii) (minusx)(minusy) = xy(iii) w(xminus y) = wxminuswy (xminus y)w = xwminus yw

In the hyperringoids though these properties are not generally valid as it can be seen in thefollowing example

Example 6 Let S be a multiplicative semigroup having a bilaterally absorbing element 0 Consider the set

P = (0 times S) cup (Stimes 0)

With the use of the hypercomposition ldquo+rdquo

(x 0) + (y 0) =(x 0) (y 0)

(0 x) + (0 y) =(0 x) (0 y)

(x 0) + (0 y) = (0 y) + (x 0) =(x 0) (0 y)

f or x y

(x 0) + (0 x) = (0 x) + (x 0) =(x 0) (0 x) (0 0)

58


P becomes a fortified join hypergroup with neutral element (0 0) If (0 x) is denoted by x and (0 0) by 0then the opposite of x is minusx = (x 0) Obviously this hypergroup has not c-elements Now let us introduce in Pa multiplication defined as follows

(x1 y1) middot (x2 y2) = (x1x2 y1y2)

This multiplication makes (P+ middot) a join hyperring in which

minus(x y) = minus[(0 x)(0 y)] = minus(0 xy) = (xy 0) 0 while x(minusy) = (0 x)(y 0) = (0 0) = 0

Similarly (minusx)y = 0 minus(x y) Furthermore

(minusx)(minusy) = (x 0)(y 0) = (xy 0) = minusxy

More examples of hyperringoids can be found in [61113ndash116]

3 Hypercompositional Algebra and Geometry

It is very well known that there exists a close relation between Algebra and Geometry So as itshould be expected this relation also appears between Hypercompositional Algebra and GeometryIt is really of exceptional interest that the axioms of the hypergroup are directly related to certainEuclidrsquos postulates [121] Indeed according to the first postulate of Euclid

ldquoHιτ


the superring and the superfield in which both the addition and the multiplication are hypercompositions and more precisely the additive part is a canonical hypergroup and the multiplicative part is a semi-hypergroup [118ndash120] In the recent bibliography a structure whose additive part is a hypergroup and the multiplicative part is a semi-group is also referred to with the term additive hyperring and similarly the term multiplicative hyperring is used when the multiplicative part is a hypergroup

Rings and Krasnerrsquos hyperrings have many common elementary algebraic properties eg in both structures the following are true

(i) ( ) ( ) x y x y xyminus = minus = minus

(ii) ( )( ) x y xyminus minus =

(iii) ( ) ( ) w x y wx wy x y w xw ywminus = minus minus = minus

In the hyperringoids though these properties are not generally valid as it can be seen in the following example


times= cup times 0 0( ) ( )P S S


( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x

P becomes a fortified join hypergroup with neutral element ( )00 If ( )0x is denoted by x and ( )00 by

0 then the opposite of x is ( )minus = 0x x Obviously this hypergroup has not c-elements Now let us

introduce in P a multiplication defined as follows

( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y

This multiplication makes + sdot( ) P a join hyperring in which

( ) ( )( ) ( ) ( ) minus = minus = minus ne =0 0 0 0 0x y x y x xyy while ( ) ( )( ) ( )minus = = =0 0 00 0x y x y

Similarly ( ) ( )0x y x yminus = ne minus Furthermore

( )( ) ( )( ) ( )minus minus = = minus=0 0 0x y xyx y xy



It is very well known that there exists a close relation between Algebra and Geometry So as it should be expected this relation also appears between Hypercompositional Algebra and Geometry It is really of exceptional interest that the axioms of the hypergroup are directly related to certain Euclidrsquos postulates [121] Indeed according to the first postulate of Euclid

ldquoΗιτήσθω ἀπό παντός σημείου ἐπί πᾶν σημεῖον εὐθεῖαν γραμμήν ἀγαγεῖνrdquo [121]

(Let the following be postulated to draw a straight line from any point to any point [122])

σθω











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y





More examples of hyperringoids can be found in [61113ndash116]ἀ





πoacute παντoacuteς σηmicroε











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










oυ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y





More examples of hyperringoids can be found in [61113ndash116] ἐ





π











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










π











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ν σηmicroε

Mathematics 2020 8 x doi FOR PEER REVIEW wwwmdpicomjournalmathematics

Article

Hypercompositional Algebra Computer Science and Geometry Gerasimos Massouros 1 and Christos Massouros 2

1 School of Social Sciences Hellenic Open University Aristotelous 18 Patra GR 26335 Greece 2 Core Department Euripus Campus National and Kapodistrian University of Athens Psahna Euboia GR

34400 Greece ChrMasuoagr or ChMassourosgmailcom Correspondence germasourosgmailcom

Received 21 May 2020 Accepted 14 July 2020 Published date

Abstract The various branches of Mathematics are not separated between themselves On the contrary they interact and extend into each otherrsquos sometimes seemingly different and unrelated areas and help them advance In this sense the Hypercompositional Algebrarsquos path has crossed among others with the paths of the theory of Formal Languages Automata and Geometry This paper presents the course of development from the hypergroup as it was initially defined in 1934 by F Marty to the hypergroups which are endowed with more axioms and allow the proof of Theorems and Propositions that generalize Kleenrsquos Theorem determine the order and the grade of the states of an automaton minimize it and describe its operation The same hypergroups lie underneath Geometry and they produce results which give as Corollaries well known named Theorems in Geometry like Hellyrsquos Theorem Kakutanirsquos Lemma Stonersquos Theorem Radonrsquos Theorem Caratheodoryrsquos Theorem and Steinitzrsquos Theorem This paper also highlights the close relationship between the hyperfields and the hypermodules to geometries like projective geometries and spherical geometries

Keywords hypergroup hyperfield formal languages automata convex set vector space geometryἀ

1 Introduction

This paper is written in the context of the special issue ldquoHypercompositional Algebra and Applicationsrdquo in ldquoMathematicsrdquo and it aims to shed light on two areas where the Hypercompositional Algebra has expanded and has interacted with them Computer Science and Geometry ῖ

Hypercompositional Algebra is a branch of Abstract Algebra which appeared in the 1930s via the introduction of the hypergroup

It is interesting that the group and the hypergroup are two algebraic structures which satisfy exactly the same axioms ie the associativity and the reproductivity but they differ in the law of synthesis In the first one the law of synthesis is a composition while in the second one it is a hypercomposition This difference makes the hypergroup a much more general algebraic structure than the group and for this reason the hypergroup has been gradually enriched with further axioms which are either more powerful or less powerful leading thus to a significant number of special hypergroups Among them there exist hypergroups that were proved to be very useful for the study of Formal Languages and Automata as well as convexity in Euclidian vector spaces Furthermore based on these hypergroups there derived other hypercompositional structures which are equally as useful in the study of Geometries (spherical projective tropical etc) and Computer Science

A binary operation (or composition) sdot on a non-void set E is a rule which assigns a unique element of E to each element of E Etimes The notation i j ka a asdot = where i j ka a a are elements of

oν ε


Article







1 Introduction

This paper is written in the context of the special issue ldquoHypercompositional Algebra and Applicationsrdquo in ldquoMathematicsrdquo and it aims to shed light on two areas where the Hypercompositional Algebra has expanded and has interacted with them Computer Science and Geometry ὐ




θε


Article







1 Introduction





αν γραmicromicro











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ν











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










γαγε


Article







1 Introduction





νrdquo [121]


So to any pair of points (a b) the segment of the straight line ab can be mapped This segmentalways exists and it is a nonempty set of points In fact it is a multivalued result of the composition oftwo elements Thus a hypercomposition has been defined in the set of the points Next according tothe second postulate

ldquoKα


Article







1 Introduction

This paper is written in the context of the special issue ldquoHypercompositional Algebra and Applicationsrdquo in ldquoMathematicsrdquo and it aims to shed light on two areas where the Hypercompositional Algebra has expanded and has interacted with them Computer Science and Geometry ὶ




πεπερασmicro


So to any pair of points ( )a b the segment of the straight line ab can be mapped This

segment always exists and it is a nonempty set of points In fact it is a multivalued result of the composition of two elements Thus a hypercomposition has been defined in the set of the points Next according to the second postulate

ldquoΚαὶ πεπερασμένην εὐθεῖαν κατά τό συνεχές ἐπrsquo εὐθείας ἐκβαλεῖνrdquo [121]

(Τo produce a finite straight line continuously in a straight line [122])

The sets a b and b a are nonempty Therefore as per Proposition 2 the reproductive axiom is valid Besides it is easy to prove that the associativity holds in the set of the points It is only necessary to keep in mind the definition of the equal figures given by Euclid in the ldquoCommon Notionsrdquo

ldquoΤὰ τῷ αὐτῷ ἲσα καὶ ἀλλήλοις ἐστίν ἲσαrdquo [121]

(Things which are equal to the same thing are also equal to one another [122])

So the set of the points is a hypergroup Moreover through similar reasoning it can be proved that any Euclidean space of dimension n can become a hypergroup Indeed

Proposition 22 Let ( )V + be α linear space over an ordered field ( )F + sdot Then V with the

hypercomposition

1|xy x y Fκ λ κ λ κ λ+= + isin + =

becomes α join hypergroup

This hypergroup is called attached hypergroup Properties of vector spaces can be found via the attached hypergroup Thus for example

Proposition 23 In α vector space V over an ordered field F the elements 1 ia i k= are affinely

dependent if and only if there exist distinct integers 1 1 n ms s t t that belong to 1 k such that

1 1

n ms s t ta a a a cap ne empty

In fact several hypergroups can be attached to a vector space [28] The connection of hypercompositional structures with Geometry was initiated by W Prenowitz [16ndash19] The classical geometries descriptive geometries spherical geometries and projective geometries can be treated as certain kinds of hypergoups all satisfying the transposition axiom The hypercomposition plays the central role in this approach It assigns the appropriate connection between any two distinct points Thus in Euclidian geometry it gives the points of the segment in spherical geometry it gives the points of the minor arc of the great circle in projective geometry it gives the point of the line This development is dimension free and it is applicable to spaces of arbitrary dimension finite or infinite

31 Hypergroups and Convexity

Several geometric notions can be described with the use of the hypercomposition One such notion is the convexity It is known that a figure is called convex if the segment joining any pair of its points lies entirely in it As mentioned above the set of the points of the plane as well as the set of the points of any vector space V over an ordered field becomes a hypergroup under the hypercomposition defined in Proposition 22 From this point of view that is with the use of the hypercomposition a subset E of V is convex if ab Esube for all a b Eisin However a subset E of a hypergroup which has this property is a semi-subhypergroup [14] Thus

νην ε


Article







1 Introduction





θε


Article







1 Introduction





αν κατ











hypercomposition






1 1





τoacute συνεχ











hypercomposition






1 1





ς











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










πrsquo ε


Article







1 Introduction





θε











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ας











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










κβαλε


Article







1 Introduction





νrdquo [121]

(To produce a finite straight line continuously in a straight line [122])

The sets a b and b a are nonempty Therefore as per Proposition 2 the reproductive axiom is validBesides it is easy to prove that the associativity holds in the set of the points It is only necessary tokeep in mind the definition of the equal figures given by Euclid in the ldquoCommon Notionsrdquo

ldquoT











hypercomposition






1 1





τ


Article







1 Introduction

This paper is written in the context of the special issue ldquoHypercompositional Algebra and Applicationsrdquo in ldquoMathematicsrdquo and it aims to shed light on two areas where the Hypercompositional Algebra has expanded and has interacted with them Computer Science and Geometry ῷ




α


Article







1 Introduction





τ


Article







1 Introduction






Article







1 Introduction

This paper is written in the context of the special issue ldquoHypercompositional Algebra and Applicationsrdquo in ldquoMathematicsrdquo and it aims to shed light on two areas where the Hypercompositional Algebra has expanded and has interacted with them Computer Science and Geometry ἲ




σα κα


Article







1 Introduction















( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










λλ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










λoις











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










στ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ν


Article







1 Introduction





σαrdquo [121]


So the set of the points is a hypergroup Moreover through similar reasoning it can be provedthat any Euclidean space of dimension n can become a hypergroup Indeed

Proposition 23 Let (V+) be α linear space over an ordered field (F+ middot) Then V with the hypercomposition

xy =κx + λy

∣∣∣ κλ isin F+ κ+ λ = 1


This hypergroup is called attached hypergroup Properties of vector spaces can be found via theattached hypergroup Thus for example

59


Proposition 24 In α vector space V over an ordered field F the elements ai i = 1 k are affinely dependentif and only if there exist distinct integers s1 sn t1 tm that belong to 1 k such that

[as1 asn ] cap [at1 atm ] empty

In fact several hypergroups can be attached to a vector space [28] The connection ofhypercompositional structures with Geometry was initiated by W Prenowitz [16ndash19] The classicalgeometries descriptive geometries spherical geometries and projective geometries can be treated ascertain kinds of hypergoups all satisfying the transposition axiom The hypercomposition plays thecentral role in this approach It assigns the appropriate connection between any two distinct pointsThus in Euclidian geometry it gives the points of the segment in spherical geometry it gives the pointsof the minor arc of the great circle in projective geometry it gives the point of the line This developmentis dimension free and it is applicable to spaces of arbitrary dimension finite or infinite


Several geometric notions can be described with the use of the hypercomposition One suchnotion is the convexity It is known that a figure is called convex if the segment joining any pair of itspoints lies entirely in it As mentioned above the set of the points of the plane as well as the set of thepoints of any vector space V over an ordered field becomes a hypergroup under the hypercompositiondefined in Proposition 23 From this point of view that is with the use of the hypercompositiona subset E of V is convex if ab sube E for all a b isin E However a subset E of a hypergroup which has thisproperty is a semi-subhypergroup [14] Thus

Proposition 25 The convex subsets of a vector space V are the semi-subhypergroups of its attached hypergroup

Consequently the properties of the convex sets of a vector space are simple applications of theproperties of the semi-subhypergroups or the subhypergroups of a hypergroup and more preciselythe attached hypergroup So this approach except from the fact that it leads to remarkable results it alsogives the opportunity to generalize the already known Theorems of the vector spaces in sets with feweraxioms than the ones of the vector spaces Next we will present some well-known named Theoremsthat arise as corollaries of more general Theorems which are valid in hypercompositional algebra

In hypergroups the following Theorem holds [214]

Theorem 5 Let H be α hypergroup in which every set with cardinality greater than n has two disjoint subsetsA B such that [A] cap [B] empty If (Yi)iisinI with card I ge n is α finite family of semi-subhypergroups of Hin which the intersection of every n elements is non-void then all the sets Yi have α non-void intersection

The combination of Propositions 24 25 and Theorem 5 gives the corollary

Corollary 3 (Hellyrsquos Theorem) Let (Ci)iisinI be a finite family of convex sets in Rd with d + 1 lt card I Thenif any d + 1 of the sets Ci have a non-empty intersection all the sets Ci have a non-empty intersection

Next the following Theorem stands for a join hypergroup

Theorem 6 Let A B be two disjoint semi-subhypergroups in a join hypergroup and let x be an idempotentelement not in the union Acup B Then [Acup x] cap B = empty or [Bcup x] capA = empty

The proof of Theorem 6 is found in [214] and it is repeated here for the purpose of demonstratingthe techniques which are used for it

60


Proof Suppose that [Acup x] cap B empty and [Bcup x] cap A empty Since x is idempotent the equalities[Acup x] = Ax and [Bcup x] = Bx are valid Thus there exist a isin A and b isin B such that ax cap B emptyand bx capA empty Hence x isin B a and x isin A b Thus B a capA b empty Next the applicationof the transposition axiom gives Bb cap Aa empty However Bb sube B and Aa sube A since A B aresemi-subhypergroups Therefore Acap B empty which contradicts the Theoremrsquos assumption

Corollary 4 Let H be a join hypergroup endowed with an open hypercomposition If A B are two disjointsemi-subhypergroups of H and x is an element not in the union Acup B then

[Acup x] cap B = empty or [Bcup x] capA = empty

The attached hypergroup of a vector space which is defined in Proposition 23 is a join hypergroupwhose hypercomposition is open so Corollary 4 applies to it and we get the Kakutanirsquos Lemma

Corollary 5 (Kakutanirsquos Lemma) If A B are disjoint convex sets in a vector space and x is a point not in theirunion then either the convex envelope of Acup x and B or the convex envelope of Bcup x and A are disjoint

Next in [2] it is proved that the following Theorem is valid

Theorem 7 Let H be a join hypergroup consisting of idempotent elements and suppose that A B are twodisjoint semi-subhypergroups in H Then there exist disjoint semi-subhypergroups M N such that A sube MB sube N and H = McupN

A direct consequence of Theorem 7 is Stonersquos Theorem

Corollary 6 (Stonersquos Theorem) If A B are disjoint convex sets in a vector space V there exist disjoint convexsets M and N such that A subeM B sube N and V = McupN

During his study of Geometry with hypercompositional structures W Prenowitz introduced theexchange spaces which are join spaces satisfying the axiom

if c isin 〈a b〉 and c a then 〈a b〉 = 〈a c〉

The above axiom enabled Prenowitz to develop a theory of linear independence and dimensionof a type familiar to classical geometry Moreover a generalization of this theory has been achieved byFreni who developed the notions of independence dimension etc in a hypergroup H that satisfiesonly the axiom

x isin langAcup

yrang

x lt 〈A〉 rArr y isin 〈Acup x〉 for every x y isin H and A sube H

Freni called these hypergroups cambiste [2526] A subset B of a hypergroup H is called free orindependent if either B = empty or x lt 〈Bminus x〉 for all x isin B otherwise it is called non-free or dependent Bgenerates H if 〈B〉 = H in which case B is a set of generators of H A free set of generators is a basis ofH Freni proved that all the bases of a cambiste hypergroup have the same cardinality The dimensionof a cambiste hypergroup H is the cardinality of any basis of H The dimension theory gives veryinteresting results in convexity hypergroups A convexity hypergroup is a join hypergroup which satisfiesthe axioms

i the hypercomposition is openii abcap ac empty implies b = c or b isin ac or c isin ab

Prenowitz defined this hypercompositional structure with equivalent axioms to the above namedit convexity space and used it as did Bryant and Webster [24] for generalizing some of the theory of

61


linear spaces In [2] it is proved that every convexity hypergroup is a cambiste hypergroup Moreoverin [2] it is proved that the following Theorem stands for convexity hypergroups

Theorem 8 Every n+1 elements of a n-dimensional convexity hypergroup H are correlated

One can easily see that the attached hypergroup of a vector space is a convexity hypergroup andmoreover if the dimension of the attached hypergroup HV of a vector space V is n then the dimensionof V is nminus 1 Thus we have the following corollary of Theorem 8

Corollary 7 (Radonrsquos Theorem) Any set of d+2 points in Rd can be partitioned into two disjoint subsetswhose convex hulls intersect

Furthermore the following Theorem is proved in [2]

Theorem 9 If x is an element of a n-dimensional convexity hypergroup H and a1 an an+1 are n+1elements of H such that x isin a1 middot middot middot anan+1 then there exists a proper subset of these elements which contains x intheir hyperproduct

A direct consequence of this Theorem is Caratheodoryrsquos Theorem

Corollary 8 (Caratheodoryrsquos Theorem) Any convex combination of points in Rd is a convex combination of atmost d+1 of them

In addition Theorems of the hypercompositional algebra are proved in [2] which give ascorollaries generalizations and extensions of Caratheodoryrsquos Theorem

An element a of a semi-subhypergroup S is called interior element of S if for each x isin S x athere exists y isin S y a such that a isin xy In [2] it is proved that any interior element of asemi-subhypergroup S of a n-dimensional convexity hypergroup is interior to a finitely generatedsemi-subhypergroup of S More precisely the following Theorem is valid [2]

Theorem 10 Let a be an interior element of a semi-subhypergroup S of a n-dimensional convexity hypergroupH Then a is interior element of a semi-subhypergroup of S which is generated by at most 2n elements

A corollary of this Theorem when H is Rd is Steinitzrsquos Theorem

Corollary 9 (Steinitzrsquos Theorem) Any point interior to the convex hull of a set E in Rd is interior to the convexhull of a subset of E containing 2d points at the most

D Freni in [123] extended the use of the hypergroup in more general geometric structures calledgeometric spaces A geometric space is a pair (S B) such that S is a non-empty set whose elements arecalled points and B is a non-empty family of subsets of S whose elements are called blocks Freni wasfollowed by S Mirvakili SM Anvariyeh and B Davvaz [124125]

32 Hyperfields and Geometry

As it is mentioned in the previous Section 22 the hyperfield was introduced by M Krasner inorder to define a certain approximation of a complete valued field by a sequence of such fields [79]The construction of this hyperfield which was named by Krasner himself residual hyperfield is alsodescribed in his paper [117]

Definition 8 A hyperring is a hypercompositional structure (H+ middot) where H is a non-empty set ldquomiddotrdquo is aninternal composition on H and ldquo+rdquo is a hypercomposition on H This structure satisfies the axioms

62


i (H+) is a canonical hypergroupii (H middot) is a multiplicative semigroup in which the zero element 0 of the canonical hypergroup is a bilaterally

absorbing elementiii the multiplication is distributive over the hypercomposition (hyperaddition) ie

z(x + y) = zx + zy and (x + y)z = xz + yz

for all x y z isin H

If H0 is a multiplicative group then (H+ middot) is called hyperfield

J Mittas studied these hypercompositional structures in a series of papers [126ndash133] Among theplenitude of examples which are found in these papers we will mention the one which is presented inthe first paragraph of [130]

Example 7 Let (Emiddot) be a totally ordered semigroup having a minimum element 0 which is bilaterally absorbingwith regards to the multiplication The following hypercomposition is defined on E

x + y =

max

x y

i f x y

z isin E | z le x i f x = y

Then (E+middot) is a hyperring If E0 is a multiplicative group then (E+middot) is a hyperfield

We referred to Mittasrsquo example because nowadays this particular hyperfield is called tropicalhyperfield (see eg [134ndash138]) and it is proved to be an appropriate and effective algebraic tool for thestudy of tropical geometry

M Krasner worked on the occurrence frequency of such structures as the hyperrings andhyperfields and he generalized his previous construction of the residual hyperfields He observedthat if R is a ring and G is a normal subgroup of Rrsquos multiplicative semigroup then the multiplicativeclasses x = xG x isin R form a partition of R and that the product of two such classes as subsets ofR is a class modG as well while their sum is a union of such classes Next he proved that the setR = R G of these classes becomes a hyperring if the product of Rlsquos two elements is defined to betheir set-wise product and their sum to be the set of the classes contained in their set-wise sum [117]

x middot y = xyG

andx + y =

zG

∣∣∣ z isin xG + yG

He also proved that if R is a field then RG is a hyperfield Krasner named thesehypercompositional structures quotient hyperring and quotient hyperfield respectively

In the recent bibliography there appear hyperfields with different and not always successfulnames all of which belong to the class of the quotient hyperfields For instance

(a) starting from the papers [139140] by A Connes and C Consani there appeared many papers(eg [135ndash138]) which gave the name laquoKrasnersrsquo hyperfieldraquo to the hyperfield which is constructedover the set 0 1 using the hypercomposition

0 + 0 = 0 0 + 1 = 1 + 0 = 1 1 + 1 = 0 1

Oleg Viro in his paper [135] is reasonably noticing that laquoTo the best of my knowledge K did notappear in Krasnerrsquos papersraquo Actually this is a quotient hyperfield Indeed let F be a field and letF be its multiplicative subgroup Then the quotient hyperfield F F = 0 F is isomorphic

63


to the hyperfield considered by A Connes and C Consani Hence the two-element non-trivialhyperfield is isomorphic to a quotient hyperfield

(b) Papers [139140] by A Connes and C Consani show the construction of the hyperfield which isnow called laquosign hyperfieldraquo in the recent bibliography over the set minus1 0 1 with the followinghypercomposition

0 + 0 = 0 0 + 1 = 1 + 0 = 1 1 + 1 = 1 minus1minus 1 = minus1 1minus 1 = minus1 + 1 = minus1 0 1

However this hyperfield is a quotient hyperfield as well Indeed let F be an ordered field and letF+ be its positive cone Then the quotient hyperfield F F+ =

minusF+ 0 F+is isomorphic to the

hyperfield which is called sign hyperfield(c) The so called laquophase hyperfieldraquo (see eg [135136]) in the recent bibliography is just the quotient

hyperfield C R+ where C is the field of complex numbers and R+ the set of the positivereal numbers The elements of this hyperfield are the rays of the complex field with originthe point (00) The sum of two elements zR+ wR+ of C R+ with zR+ wR+ is the set(zp + wq)R+

∣∣∣ p q isin R+ which consists of all the interior rays xR+ of the convex angle which is

created from these two elements while the sum of two opposite elements gives the participatingelements and the zero element This hyperfield is presented in detail in [141]

Krasner immediately realized that if all hyperrings could be isomorphically embedded intoquotient hyperrings then several conclusions of their theory could be deduced in a very straightforwardmanner through the use of the ring theory So he raised the question whether all the hyperringsare isomorphic to subhyperrings of quotient hyperrings or not He also raised a similar questionregarding the hyperfields [117] These questions were answered by C Massouros [142ndash144] and thenby A Nakassis [145] via the following Theorems

Theorem 11 [142143] Let (Θ middot) be a multiplicative group Let H = Θ cup 0 where 0 is a multiplicativelyabsorbing element If H is equipped with the hypercomposition

x + 0 = 0 + x = x f or all x isin Hx + x = Hx f or all x isin Θx + y = y + x =

x y

f or all x y isin Θ with x y

then (H+ middot) is a hyperfield which does not belong to the class of quotient hyperfields when Θ is a periodic group

Theorem 12 [144] Let Θ = Θ otimes 1minus1 be the direct product of the multiplicative groups Θ and minus1 1where card Θ gt 2 Moreover let K = Θ cup 0 be the union of Θ with the multiplicatively absorbing element 0If K is equipped with the hypercomposition

w + 0 = 0 + w = w f or all w isin K(x i) + (x i) = K

(x i) (xminusi) 0

f or all (x i) isin Θ

(x i) + (xminusi) = K(x i) (xminusi)


(x i) + (y j) =(x i) (xminusi) (y j) (yminus j)

f or all (x i) (y j) isin Θ with (y j) (x i) (xminusi)

then (K+ middot) is a hyperfield which does not belong to the class of quotient hyperfields when Θ is a periodic group

Proposition 26 [145] Let (T middot) be a multiplicative group of m m gt 3 elements Let T = Tcup 0 where 0 isa multiplicatively absorbing element If T is equipped with the hypercomposition

a + 0 = 0 + a = a f or all a isin Ta + a = 0 a f or all a isin Ta + b = b + a = T0 a b f or all a b isin T with a b

64


then (T+ middot) is a hyperfield

Theorem 13 [145] If T is a finite multiplicative group of m m gt 3 elements and if the hyperfield Tis isomorphic to a quotient hyperfield F Q then Q cup 0 is a field of m-1 elements while F is a field of(mminus 1)2 elements

Clearly we can choose the cardinality of T in such a way that T cannot be isomorphic to a quotienthyperfield In [144145] one can find non-quotient hyperrings as well

Therefore we know 4 different classes of hyperfields so far the class of the quotient hyperfieldsand the three ones which are constructed via the Theorems 11 12 and 13

The open and closed hypercompositions [5] in the hyperfields are of special interest Regardingthese we have the following

Proposition 27 In a hyperfield K the sum x + y of any two non-opposite elements x y 0 does not contain theparticipating elements if and only if the difference xminus x equals to minusx 0 x for every x 0

Proposition 28 In a hyperfield K the sum x + y of any two non-opposite elements x y 0 contains these twoelements if and only if the difference xminus x equals to H for every x 0

For the proofs of the above Propositions 27 and 28 see [141] With regard to Proposition 28 it isworth mentioning that there exist hyperfields in which the sum x + y contains only the two addendsx y ie x + y =

x y

when y minusx and x y 0 [141]

Theorem 14 [141] Let (K+ middot) be a hyperfield Let dagger be a hypercomposition on K defined as follows

x dagger y = (x + y)cup x y

i f y minusx and x y 0

x dagger (minusx) = K f or all x isin K0x dagger 0 = 0 dagger x = x f or all x isin K

Then (K dagger middot) is a hyperfield and moreover if (K+ middot) is a quotient hyperfield then (K dagger middot) is a quotienthyperfield as well

Corollary 10 If (K+ middot) is a field then (K dagger middot) is a quotient hyperfield

The following problem in field theory is raised from the study of the isomorphism of the quotienthyperfields to the hyperfields which are constructed with the process given in Theorem 14

when does a subgroup G of the multiplicative group of a field F have the ability to generate F via thesubtraction of G from itself [141143]

A partial answer to this problem which is available so far regarding the finite fields is given with thefollowing theorem

Theorem 15 [146] Let F be a finite field and G be a subgroup of its multiplicative group of indexn and order mThen G-G = F if and only if

n = 2 and m gt 2n = 3 and m gt 5n = 4 minus1 isin G and m gt 11n = 4 minus1 lt G and m gt 3n = 5 charF = 2 and m gt 8n = 5 charF = 3 and m gt 9n = 5 charF 2 3 and m gt 23

65


Closely related to the hyperfield is the hypermodule and the vector space

Definition 9 A left hypermodule over a unitary hyperring P is a canonical hypergroup M with an externalcomposition (a m)rarr am from PtimesM to M satisfying the conditions

i a(m + n) = am + anii (a + b)m = am + bmiii (ab)m = a(bm)iv 1m = m and 0m = 0

for all a b isin P and all m n isinM

The right hypermodule is defined in a similar way A hypermodule over a hyperfield is calledvector hyperspace

Suppose V and W are hypermodules over the hyperring P The cartesian product V timesWcan become a hypercompositional structure over P when the operation and the hyperoperationfor v v1 v2 isin V w w1 w2 isinW and α isin P are defined componentwise as follows

(v1 w1) + (v2 w2) =⋃

(v w)∣∣∣ v isin v1 + v2 w isin w1 + w2

a(v w) = (av aw)

The resulting hypercompositional structure is called the direct sum of V and W

Theorem 16 The direct sum of the hypermodules is not a hypermodule

Proof Let V and W be two hypermodules over a hyperring P Then

(a + b)(v w) =⋃

c(v w)∣∣∣ c isin a + b

=

⋃(cv cw)

∣∣∣ c isin a + b

On the other hand

a(v w) + b(v w) = (av aw) + (bv bw) =⋃

(x y)∣∣∣ x isin av + bv y isin aw + bw

=

=⋃

(x y)∣∣∣ x isin (a + b)v y isin (a + b)w

=

⋃(sv rw)

∣∣∣ s r isin a + b

Therefore(a + b)(v w) sube a(v w) + b(v w)

Consequently axiom (ii) is not valid

Remark 2 Errors in Published Papers Unfortunately there exist plenty of papers which incorrectly considerthat the direct sum of hypermodules is a hypermodule For instance they mistakenly consider that if P is ahyperring or a hyperfield then P n is a hypermodule or a vector hyperspace over P respectively Due to this errora lot of if not all the conclusions of certain papers are incorrect We are not going to specifically refer to suchpapers as we do not wish to add negative citations in our paper but we refer positively to the paper by P AmeriM Eyvazi and S Hoskova-Mayerova [120] where the authors have presented a counterexample which shows thatthe polynomials over a hyperring give a superring in the sense of Mittas [118119] and not a hyperring as it ismistakenly mentioned in a previously published paper which is referred there This error can also be highlightedwith the same method as the one in Theorem 16 since the polynomials over a hyperring P can be considered asthe ordered sets (a0 a1 ) where ai i=0 1 are their coefficients

Following the above remark we can naturally introduce the definition

66


Definition 10 A left weak hypermodule over a unitary hyperring P is a canonical hypergroup M with anexternal composition (a m)rarr am from PtimesM to M satisfying the conditions (i) (iii) (iv) of the Definition 9and in place of (ii) the condition

iirsquo (a + b)m sube am + bm for all a b isin P and all m isinM

The quotient hypermodule over a quotient hyperring is constructed in [147] as followsLet M be a Pminusmodule where P is a unitary ring and let G be a subgroup of the multiplicative

semigroup of P which satisfies the condition aG bG = abG for all a b isin P Note that this condition isequivalent to the normality of G only when P0 is a group which appears only in the case of divisionrings (see [144]) Next we introduce in M the following equivalence relation

x sim yhArr x = ty t isin G

After that we equip M with the following hypercomposition where M is the set of equivalence classesof M modulo ~

x


P is a hyperring or a hyperfield then nP is a hypermodule or a vector hyperspace over P respectively Due to this error a lot of if not all the conclusions of certain papers are incorrect We are not going to specifically refer to such papers as we do not wish to add negative citations in our paper but we refer positively to the paper by Ρ Ameri M Eyvazi and S Hoskova-Mayerova [120] where the authors have presented a counterexample which shows that the polynomials over a hyperring give a superring in the sense of Mittas [118 119] and not a hyperring as it is mistakenly mentioned in a previously published paper which is referred there This error can also be highlighted with the same method as the one in Theorem 16 since the polynomials over a hyperring P can be considered as the ordered sets ( )0 1 a a where ia i=0 1 hellip are their coefficients


Definition 10 A left weak hypermodule over a unitary hyperring P is a canonical hypergroup M with an external composition ( )a m amrarr from P Mtimes to M satisfying the conditions (i) (iii) (iv) of the

Definition 9 and in place of (ii) the condition

iirsquo ( ) a b m am bm+ sube + for all a b Pisin and all m Misin

The quotient hypermodule over a quotient hyperring is constructed in [147] as follows Let M be a P minus module where P is a unitary ring and let G be a subgroup of the multiplicative semigroup of P which satisfies the condition aG bG abG= for all a b Pisin Note

that this condition is equivalent to the normality of G only when 0P is a group which appears

only in the case of division rings (see [144]) Next we introduce in M the following equivalence relation

~ x y x ty t GhArr = isin

After that we equip M with the following hypercomposition where M is the set of equivalence classes of M modulo ~

+ = isin = + isin

| x y w M w xp yq p q G

ie x y+

consists of all the classes w Misin which are contained in the set-wise sum of x y Then

( ) +M becomes a canonical hypergroup Let P be the quotient hyperring of P over G We

consider the external composition from P Mtimes to M defined as follows

a x ax= for each a P x Misin isin

This composition satisfies the axioms of the hypermodule and so M becomes a P - hypermodule If M is a module over a division ring D then using the multiplicative group D of D we

can construct the quotient hyperring 0 D D D D= = and the relevant quotient hypermodule

M For any a Misin it holds that 0a a a+ = In [147] it is shown that this hypermodule is

strongly related to the projective geometries A Connes and C Consani in [139140] also prove that the projective geometries in which the lines have at least four points are exactly vector hyperspaces over the quotient hyperfield with two elements Moreover if V is a vector space over an ordered field F then using the positive cone F+ of F we can construct the vector hyperspace V over the

quotient hyperfield 0 F F F F Fminus ++= = In [147] it is shown that every Euclidean spherical

geometry can be considered as a quotient vector hyperspace over the quotient hyperfield with three elements

Modern algebraic geometry is based on abstract algebra which offers its techniques for the study of geometrical problems In this sense the hyperfields were connected to the conic sections

y =w isinM

∣∣∣ w = xp + yq p q isin G

ie x












+ = isin = + isin


ie x y+












y consists of all the classes w isinM which are contained in the set-wise sum of x y Then(M












+ = isin = + isin


ie x y+












)

becomes a canonical hypergroup Let P be the quotient hyperring of P over G We consider the externalcomposition from PtimesM to M defined as follows

a x = ax for each a isin P x isinM

This composition satisfies the axioms of the hypermodule and so M becomes a P- hypermoduleIf M is a module over a division ring D then using the multiplicative group D of D we can

construct the quotient hyperring D = D D = 0 D and the relevant quotient hypermodule MFor any a isin M it holds that a + a =

0 a

In [147] it is shown that this hypermodule is strongly

related to the projective geometries A Connes and C Consani in [139140] also prove that theprojective geometries in which the lines have at least four points are exactly vector hyperspaces overthe quotient hyperfield with two elements Moreover if V is a vector space over an ordered fieldF then using the positive cone F+ of F we can construct the vector hyperspace V over the quotienthyperfield F = F F+ =

Fminus 0 F+

In [147] it is shown that every Euclidean spherical geometrycan be considered as a quotient vector hyperspace over the quotient hyperfield with three elements

Modern algebraic geometry is based on abstract algebra which offers its techniques for the studyof geometrical problems In this sense the hyperfields were connected to the conic sections via anumber of papers [148ndash150] where the definition of an elliptic curve over a field F was naturallyextended to the definition of an elliptic hypercurve over a quotient Krasner hyperfield The conclusionsobtained in [148ndash150] were extended to cryptography as well


In this paper we have initially presented the relationship between the groups and the hypergroupsIt is interesting that the groups and the hypergroups are two algebraic structures which satisfy exactlythe same axioms ie the associativity and the reproductivity but they differ in the law of synthesisIn the first ones the law of synthesis is a composition while in the second ones it is a hypercompositionThis difference makes the hypergroups much more general algebraic structures than the groups and forthis reason the hypergroups have been gradually enriched with further axioms which are either morepowerful or less powerful and they lead to a significant number of special hypergroups

We have also presented the connection of the Hypercompositional Algebra to the Formal Languagesand Automata theory as well as its close relationship to Geometry It is very interesting that thetransposition hypergroup which appears in the Formal Languages is the proper algebraic tool for thestudy of the convexity in Geometry The study of this hypergroup has led to general Theorems which

67


have as corollaries well known named Theorems in the vector spaces Different types of transpositionhypergroups as for example the fortified transposition hypergroup give birth to hypercompositionalstructures like the hyperringoid the linguistic hyperringoid the join hyperring the algebraic structureof which is an area with a plentitude of hitherto open problems Moreover the hyperfield and thehypermodule describe fully and accurately the projective and the spherical geometries while they aredirectly connected to other geometries as well Moreover the classification problem of hyperfieldsgives birth to the question

when does a subgroup G of the multiplicative group of a field F have the ability to generate F via thesubtraction of G from itself

This question is answered for certain finite fields only and still remains to be answered in its entirety

Author Contributions Both authors contributed equally to this work All authors have read and agreed to thepublished version of the manuscript

Funding This research received no external funding The APC was funded by Mathematics MDPI


References

1 Massouros CG Some properties of certain subhypergroups Ratio Math 2013 25 67ndash762 Massouros CG On connections between vector spaces and hypercompositional structures Ital J Pure

Appl Math 2015 34 133ndash1503 Massouros CG Massouros GG The transposition axiom in hypercompositional structures Ratio Math

2011 21 75ndash904 Massouros CG Massouros GG On certain fundamental properties of hypergroups and fuzzy

hypergroupsmdashMimic fuzzy hypergroups Internat J Risk Theory 2012 2 71ndash825 Massouros CG Massouros GG On open and closed hypercompositions AIP Conf Proc 2017 1978 340002

[CrossRef]6 Massouros CG Dramalidis A Transposition Hv-groups Ars Comb 2012 106 143ndash1607 Novaacutek M Krehliacutek Š EL-hyperstructures revisited Soft Comput 2018 22 7269ndash7280 [CrossRef]8 Marty F Sur une geacuteneacuteralisation de la notion de groupe Huitiegraveme Congregraves des matheacutematiciens Scand

Stockholm 1934 45ndash499 Marty F Rocircle de la notion de hypergroupe dans lrsquoeacutetude de groupes non abeacuteliens CR Acad Sci 1935 201

636ndash63810 Marty F Sur les groupes et hypergroupes attacheacutes agrave une fraction rationelle Ann Lrsquoeacutecole Norm 1936 53

83ndash123 [CrossRef]11 Massouros CG Massouros GG Identities in multivalued algebraic structures AIP Conf Proc 2010 1281

2065ndash2068 [CrossRef]12 Massouros CG Massouros GG On the algebraic structure of transposition hypergroups with idempotent

identity Iran J Math Sci Inform 2013 8 57ndash74 [CrossRef]13 Massouros CG Massouros GG Transposition hypergroups with idempotent identity Int J Algebr

Hyperstruct Appl 2014 1 15ndash2714 Massouros CG On the semi-subhypergroups of a hypergroup Int J Math Math Sci 1985 8 725ndash728

[CrossRef]15 Jantosciak J Transposition hypergroups Noncommutative Join Spaces J Algebra 1997 187 97ndash119

[CrossRef]16 Prenowitz W Projective Geometries as multigroups Am J Math 1943 65 235ndash256 [CrossRef]17 Prenowitz W Descriptive Geometries as multigroups Trans Am Math Soc 1946 59 333ndash380 [CrossRef]18 Prenowitz W Spherical Geometries and mutigroups Can J Math 1950 2 100ndash119 [CrossRef]19 Prenowitz W A Contemporary Approach to Classical Geometry Am Math Month 1961 68 1ndash67 [CrossRef]20 Prenowitz W Jantosciak J Geometries and Join Spaces J Reine Angew Math 1972 257 100ndash12821 Prenowitz W Jantosciak J Join Geometries A Theory of Convex Sets and Linear Geometry Springer

BerlinHeidelberg Germany 1979

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22 Jantosciak J Classical geometries as hypergroups In Proceedings of the Atti Convegno Ipergruppi AltreStructure Multivoche e Loro Applicazioni Udine Italia 15ndash18 October 1985 pp 93ndash104

23 Jantosciak J A brif survey of the theory of join spaces In Proceedings of the 5th International Congress onAHA Iasi Rommania 4ndash10 July 1993 Hadronic Press Palm Harbor FA USA 1994 pp 109ndash122

24 Bryant VW Webster RJ Generalizations of the Theorems of Radon Helly and Caratheodory Monatsh Math1969 73 309ndash315 [CrossRef]

25 Freni D Sur les hypergroupes cambistes Rend Ist Lomb 1985 119 175ndash18626 Freni D Sur la theacuteorie de la dimension dans les hypergroupes Acta Univ Carol Math Phys 1986 27 67ndash8027 Massouros CG Hypergroups and convexity Riv Mat Pura Appl 1989 4 7ndash2628 Mittas J Massouros CG Hypergroups defined from linear spaces Bull Greek Math Soc 1989 30 63ndash7829 Massouros CG Hypergroups and Geometry Mem Acad Romana Math Spec Issue 1996 19 185ndash19130 Dramalidis A On geometrical hyperstructures of finite order Ratio Math 2011 21 43ndash5831 Dramalidis A Some geometrical P-HV-hyperstructures In New Frontiers in Hyperstructures Hadronic Press

Palm Harbor FL USA 1996 pp 93ndash10232 Cristea I Complete hypergroups 1-hypergroups and fuzzy sets An Stalele Univ Ovidius Constanta 2000

10 25ndash3833 Cristea I A property of the connection between fuzzy sets and hypergroupoids Ital J Pure Appl Math

2007 21 73ndash8234 Stefanescu M Cristea I On the fuzzy grade of hypergroups Fuzzy Sets Syst 2008 159 1097ndash1106

[CrossRef]35 Corsini P Cristea I Fuzzy sets and non complete 1-hypergroups An Stalele Univ Ovidius Constanta 2005

13 27ndash5436 Corsini P Graphs and Join Spaces J Comb Inf Syst Sci 1991 16 313ndash31837 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Udine Italy 199338 Corsini P Join Spaces Power Sets Fuzzy Sets In Proceedings of the 5th International Congress on Algebraic

Hyperstructures and Applications Iasi Romania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA1994 pp 45ndash52

39 Corsini P Binary relations interval structures and join spaces J Appl Math Comput 2002 10 209ndash216[CrossRef]


41 Corsini P Leoreanu V Applications of Hyperstructures Theory Kluwer Academic Publishers BerlinGermany 2003

42 Corsini P Leoreanu-Fotea V On the grade of a sequence of fuzzy sets and join spaces determined by ahypergraph Southeast Asian Bull Math 2010 34 231ndash242

43 Leoreanu V On the heart of join spaces and of regular hypergroups Riv Mat Pura Appl 1995 17 133ndash14244 Leoreanu V Direct limit and inverse limit of join spaces associated with fuzzy sets PUMA 2000 11 509ndash51645 Leoreanu-Fotea V Fuzzy join n-ary spaces and fuzzy canonical n-ary hypergroups Fuzzy Sets Syst 2010

161 3166ndash3173 [CrossRef]46 Leoreanu-Fotea V Rosenberg I Join Spaces Determined by Lattices J Mult-Valued Logic Soft Comput

2010 16 7ndash1647 Hoskova S Abelization of join spaces of affine transformations of ordered field with proximity

Appl Gen Topol 2005 6 57ndash65 [CrossRef]48 Chvalina J Hoskova S Modelling of join spaces with proximities by first-order linear partial differential

operators Ital J Pure Appl Math 2007 21 177ndash19049 Hoskova S Chvalina J Rackova P Transposition hypergroups of Fredholm integral operators and related

hyperstructures (Part I) J Basic Sci 2006 3 11ndash1750 Hoskova S Chvalina J Rackova P Transposition hypergroups of Fredholm integral operators and related

hyperstructures (Part II) J Basic Sci 2008 4 55ndash6051 Chvalina J Hoskova-Mayerova S On Certain Proximities and Preorderings on the Transposition

Hypergroups of Linear First-Order Partial Differential Operators An Stalele Univ Ovidius ConstantaSer Mat 2014 22 85ndash103 [CrossRef]

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52 Chvalina J Chvalinova L Transposition hypergroups formed by transformation operators on rings ofdifferentiable functions Ital J Pure Appl Math 2004 15 93ndash106

53 Jantosciak J Massouros CG Strong Identities and fortification in Transposition hypergroups J DiscretMath Sci Cryptogr 2003 6 169ndash193 [CrossRef]

54 Massouros CG Canonical and Join Hypergroups An Sti Univ Al I Cuza Iasi 1996 1 175ndash18655 Massouros CG Normal homomorphisms of fortified join hypergroups Algebraic Hyperstructures and

Applications In Proceedings of the 5th International Congress Iasi Romania 4ndash10 July 1993 HadronicPress Palm Harbor FL USA 1994 pp 133ndash142

56 Massouros CG Isomorphism Theorems in fortified transposition hypergroups AIP Conf Proc 2013 15582059ndash2062 [CrossRef]

57 Massouros GG Massouros CG Mittas ID Fortified join hypergroups Ann Math Blaise Pascal 1996 3155ndash169 [CrossRef]

58 Massouros GG Zafiropoulos FA Massouros CG Transposition polysymmetrical hypergroups AlgebraicHyperstructures and Applications In Proceedings of the 8th International Congress Samothraki Greece1ndash9 September 2002 Spanidis Press Xanthi Greece 2003 pp 91ndash202

59 Massouros CG Massouros GG Transposition polysymmetrical hypergroups with strong identity J BasicSci 2008 4 85ndash93

60 Massouros CG Massouros GG On subhypergroups of fortified transposition hypergroups AIP Conf Proc2013 1558 2055ndash2058 [CrossRef]

61 Massouros GG Fortified join hypergroups and join hyperrings An Sti Univ Al I Cuza Iasi Sect I Mat1995 1 37ndash44

62 Massouros GG The subhypergroups of the fortified join hypergroup Italian J Pure Appl Math 1997 251ndash63

63 Ameri R Zahedi MM Hypergroup and join space induced by a fuzzy subset Pure Math Appl 1997 8155ndash168

64 Ameri R Topological transposition hypergroups Italian J Pure Appl Math 2003 13 181ndash18665 Ameri R Fuzzy transposition hypergroups Italian J Pure Appl Math 2005 18 147ndash15466 Rosenberg IG Hypergroups and join spaces determined by relations Italian J Pure Appl Math 1998 4

93ndash10167 Massouros GG Mittas JD LanguagesmdashAutomata and hypercompositional structures Algebraic

Hyperstructures and Applications In Proceedings of the 4th International Congress Xanthi Greece27ndash30 June 1990 World Scientific Singapore 1991 pp 137ndash147

68 Massouros GG Automata Languages and Hypercompositional Structures PhD Thesis National TechnicalUniversity of Athens Athens Greece 1993

69 Massouros GG Automata and hypermoduloids Algebraic Hyperstructures and ApplicationsIn Proceedings of the 5th International Congress Iasi Romania 4ndash10 July 1993 Hadronic Press Palm HarborFL USA 1994 pp 251ndash265

70 Massouros GG An automaton during its operation Algebraic Hyperstructures and ApplicationsIn Proceedings of the 5th International Congress Iasi Romania 4ndash10 July 1993 Hadronic Press Palm HarborFA USA 1994 pp 267ndash276

71 Massouros GG On the attached hypergroups of the order of an automaton J Discret Math Sci Cryptogr2003 6 207ndash215 [CrossRef]

72 Massouros GG Hypercompositional structures in the theory of languages and automata An Sti Univ AlI Cuza Iasi Sect Inform 1994 3 65ndash73

73 Massouros GG Hypercompositional structures from the computer theory Ratio Math 1999 13 37ndash4274 Bonansinga P Sugli ipergruppi quasicanonici Atti Soc Pelor Sc Fis Mat Nat 1981 27 9ndash1775 Massouros CG Quasicanonical hypergroups In Proceedings of the 4th Internat Cong on Algebraic

Hyperstructures and Applications Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991pp 129ndash136

76 Ioulidis S Polygroups et certaines de leurs proprieacutetieacutes Bull Greek Math Soc 1981 22 95ndash10477 Comer S Polygroups derived from cogroups J Algebra 1984 89 397ndash405 [CrossRef]78 Davvaz B Polygroup Theory and Related Systems Publisher World Scientific Singapore 2013

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79 Krasner M Approximation des corps valueacutes complets de caracteacuteristique p0 par ceux de caracteacuteristique 0 Colloque drsquoAlgegravebre Supeacuterieure (Bruxelles Decembre 1956) Centre Belge de Recherches Matheacutematiques EacutetablissementsCeuterick Louvain Librairie Gauthier-Villars Paris France 1957 pp 129ndash206

80 Mittas J Hypergroupes canoniques hypervalues CR Acad Sci Paris 1970 271 4ndash781 Mittas J Hypergroupes canoniques Math Balk 1972 2 165ndash17982 Krasner M La loi de Jordan-Holder dans les hypergroupes et les suites generatrices des corps de nombres

P-adiqes Duke Math J 1940 6 120ndash140 [CrossRef]83 Krasner M Cours drsquo Algebre superieure Theories des valuation et de Galois Cours Fac Sci Lrsquo Univ Paris

1967 68 1ndash30584 Sureau Y Sous-hypergroupe engendre par deux sous-hypergroupes et sous-hypergroupe ultra-clos drsquoun

hypergroupe CR Acad Sc Paris 1977 284 983ndash98485 Kleene SC Representation of Events in Nerve Nets and Finite Automata In Automata Studies Shannon CE

McCarthy J Eds Princeton University Princeton NJ USA 1956 pp 3ndash4286 McNaughton R Yamada H Regular Expressions and State Graphs for Automata IEEE Trans Electron

Comput 1960 9 39ndash47 [CrossRef]87 Brzozowski JA A Survey of Regular Expressions and their Applications IEEE Trans Electron Comput

1962 11 324ndash335 [CrossRef]88 Brzozowski JA Derivatives of Regular Expressions J ACM 1964 11 481ndash494 [CrossRef]89 Brzozowski JA McCluskey EJ Signal Flow Graph Techniques for Sequential Circuit State Diagrams

IEEE Trans Electron Comput 1963 EC-12 67ndash76 [CrossRef]90 Chvalina J Chvalinova L State hypergroups of Automata Acta Math Inform Univ Ostrav 1996 4

105ndash12091 Chvalina J Krehliacutek S Novaacutek M Cartesian composition and the problem of generalizing the MAC

condition to quasi-multiautomata An Stalele Univ Ovidius Constanta 2016 24 79ndash100 [CrossRef]92 Chvalina J Novaacutek M Krehliacutek S Hyperstructure generalizations of quasi-automata induced by modelling

functions and signal processing AIP Conf Proc 2019 2116 310006 [CrossRef]93 Novaacutek M Krehliacutek S Stanek D n-ary Cartesian composition of automata Soft Comput 2019 24 1837ndash1849

[CrossRef]94 Novaacutek M Some remarks on constructions of strongly connected multiautomata with the input

semihypergroup being a centralizer of certain transformation operators J Appl Math 2008 I 65ndash7295 Chorani M Zahedi MM Some hypergroups induced by tree automata Aust J Basic Appl Sci 2012 6

680ndash69296 Chorani M State hyperstructures of tree automata based on lattice-valued logic RAIRO-Theor Inf Appl

2018 52 23ndash4297 Corsini P Binary relations and hypergroupoids Italian J Pure Appl Math 2000 7 11ndash1898 Corsini P Hypergraphs and hypergroups Algebra Universalis 1996 35 548ndash555 [CrossRef]99 Gionfriddo M Hypergroups Associated with Multihomomorphisms between Generalized Graphs Corsini P Ed

Convegno su Sistemi Binary e Loro Applicazioni Taormina Italy 1978 pp 161ndash174100 Nieminen J Join Space Graph J Geom 1988 33 99ndash103 [CrossRef]101 Nieminen J Chordal Graphs and Join Spaces J Geom 1989 34 146ndash151 [CrossRef]102 De Salvo M Lo Faro G Hypergroups and binary relations Mult Valued Log 2002 8 645ndash657103 De Salvo M Lo Faro G A new class of hypergroupoids associated to binary relations J Mult Valued Logic

Soft Comput 2003 9 361ndash375104 De Salvo M Freni D Lo Faro G Fully simple semihypergroups J Algebra 2014 399 358ndash377 [CrossRef]105 Cristea I Jafarpour M Mousavi SS Soleymani A Enumeration of Rosenberg hypergroups

Comput Math Appl 2011 32 72ndash81 [CrossRef]106 Cristea I Stefanescu M Binary relations and reduced hypergroups Discrete Math 2008 308 3537ndash3544

[CrossRef]107 Cristea I Stefanescu M Angheluta C About the fundamental relations defined on the hypergroupoids

associated with binary relations Electron J Combin 2011 32 72ndash81 [CrossRef]108 Aghabozorgi H Jafarpour M Dolatabadi M Cristea I An algorithm to compute the number of Rosenberg

hypergroups of order less than 7 Ital J Pure Appl Math 2019 42 262ndash270

71



110 Massouros CG On path hypercompositions in graphs and automata MATEC Web Conf 2016 41 5003[CrossRef]

111 Massouros CG Tsitouras CG Enumeration of hypercompositional structures defined by binary relationsItal J Pure Appl Math 2011 28 43ndash54

112 Tsitouras CG Massouros CG Enumeration of Rosenberg type hypercompositional structures defined bybinary relations Eur J Comb 2012 33 1777ndash1786 [CrossRef]

113 Massouros GG The hyperringoid Mult Valued Logic 1998 3 217ndash234114 Massouros GG Massouros CG Homomorphic relations on Hyperringoids and Join Hyperrings Ratio Mat

1999 13 61ndash70115 Massouros GG Solving equations and systems in the environment of a hyperringoid Algebraic

Hyperstructures and Applications In Proceedings of the 6th International Congress Prague Czech Republic1ndash9 September 1996 Democritus Univ of Thrace Press Komotini Greece 1997 pp 103ndash113

116 Massouros CG Massouros GG On join hyperrings Algebraic Hyperstructures and ApplicationsIn Proceedings of the 10th International Congress Brno Czech Republic 3ndash9 September 1996 pp 203ndash215

117 Krasner M A class of hyperrings and hyperfields Int J Math Math Sci 1983 6 307ndash312 [CrossRef]118 Mittas J Sur certaines classes de structures hypercompositionnelles Proc Acad Athens 1973 48 298ndash318119 Mittas J Sur les structures hypercompositionnelles Algebraic Hyperstructures and Applications

In Proceedings of the 4th International Congress Xanthi Greece 27ndash30 June 1990 World ScientificSingapore 1991 pp 9ndash31

120 Ameri R Eyvazi M Hoskova-Mayerova S Superring of Polynomials over a Hyperring Mathematics 20197 902 [CrossRef]

121 Eυκλε











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










δης (Euclid) Στoιχε











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










α (Elements) c 300 BC Heiberg I Stamatis E Eds OE∆B AthensGreece 1975

122 Heath TL The Thirteen Books of Euclidrsquos Elements 2nd ed Dover Publications New York NY USA 1956123 Freni D Strongly Transitive Geometric Spaces Applications to Hypergroups and Semigroups Theory

Commun Algebra 2004 32 969ndash988 [CrossRef]124 Mirvakili S Davvaz B Strongly transitive geometric spaces Applications to hyperrings Revista UMA

2012 53 43ndash53125 Anvariyeh SM Davvaz B Strongly transitive geometric spaces associated with (m n)-ary hypermodules

An Sti Univ Al I Cuza Iasi 2013 59 85ndash101 [CrossRef]126 Mittas J Hyperanneaux et certaines de leurs proprieacuteteacutes CR Acad Sci Paris 1969 269 623ndash626127 Mittas J Hypergroupes et hyperanneaux polysymeacutetriques CR Acad Sci Paris 1970 271 920ndash923128 Mittas J Contributions a la theacuteorie des hypergroupes hyperanneaux et les hypercorps hypervalues CR

Acad Sci Paris 1971 272 3ndash6129 Mittas J Certains hypercorps et hyperanneaux deacutefinis agrave partir de corps et anneaux ordonneacutes Bull Math

Soc Sci Math R S Roum 1971 15 371ndash378130 Mittas J Sur les hyperanneaux et les hypercorps Math Balk 1973 3 368ndash382131 Mittas J Contribution agrave la theacuteorie des structures ordonneacutees et des structures valueacutees Proc Acad Athens

1973 48 318ndash331132 Mittas J Hypercorps totalement ordonnes Sci Ann Polytech Sch Univ Thessalon 1974 6 49ndash64133 Mittas J Espaces vectoriels sur un hypercorps Introduction des hyperspaces affines et Euclidiens Math Balk

1975 5 199ndash211134 Viro O On basic concepts of tropical geometry Proc Steklov Inst Math 2011 273 252ndash282 [CrossRef]135 Viro O Hyperfields for Tropical Geometry I Hyperfields and dequantization arXiv 2010 arXiv10063034136 Baker M Bowler N Matroids over hyperfields arXiv 2017 arXiv160101204137 Jun J Geometry of hyperfields arXiv 2017 arXiv170709348138 Lorscheid O Tropical geometry over the tropical hyperfield arXiv 2019 arXiv190701037139 Connes A Consani C The hyperring of adegravele classes J Number Theory 2011 131 159ndash194 [CrossRef]140 Connes A Consani C From monoids to hyperstructures In search of an absolute arithmetic arXiv 2010

arXiv10064810141 Massouros CG Constructions of hyperfields Math Balk 1991 5 250ndash257

72


142 Massouros CG Methods of constructing hyperfields Int J Math Math Sci 1985 8 725ndash728 [CrossRef]143 Massouros CG A class of hyperfields and a problem in the theory of fields Math Montisnigri 1993 1 73ndash84144 Massouros CG On the theory of hyperrings and hyperfields Algebra Logika 1985 24 728ndash742 [CrossRef]145 Nakassis A Recent results in hyperring and hyperfield theory Int J Math Math Sci 1988 11 209ndash220

[CrossRef]146 Massouros CG A Field Theory Problem Relating to Questions in Hyperfield Theory AIP Conf Proc 2011

1389 1852ndash1855 [CrossRef]147 Massouros CG Free and cyclic hypermodules Ann Mat Pura Appl 1988 150 153ndash166 [CrossRef]148 Vahedi V Jafarpour M Aghabozorgi H Cristea I Extension of elliptic curves on Krasner hyperfields

Comm Algebra 2019 47 4806ndash4823 [CrossRef]149 Vahedi V Jafarpour M Cristea I Hyperhomographies on Krasner Hyperfields Symmetry 2019 11 1442

[CrossRef]150 Vahedi V Jafarpour M Hoskova-Mayerova S Aghabozorgi H Leoreanu-Fotea V Bekesiene S Derived

Hyperstructures from Hyperconics Mathematics 2020 8 429 [CrossRef]

copy 2020 by the authors Licensee MDPI Basel Switzerland This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (httpcreativecommonsorglicensesby40)

73

mathematics

Article

Fuzzy Reduced Hypergroups

Milica Kankaras 1 and Irina Cristea 21 Department of Mathematics University of Montenegro 81000 Podgorica Montenegro milicakucgacme2 Centre for Information Technologies and Applied Mathematics University of Nova Gorica

5000 Nova Gorica Slovenia Correspondence irinacristeaungsi or irinacriyahoocouk Tel +386-0533-15-395

Received 10 January 2020 Accepted 14 February 2020 Published 17 February 2020

Abstract The fuzzyfication of hypercompositional structures has developed in several directions In thisnote we follow one direction and extend the classical concept of reducibility in hypergroups to the fuzzycase In particular we define and study the fuzzy reduced hypergroups New fundamental relationsare defined on a crisp hypergroup endowed with a fuzzy set that lead to the concept of fuzzy reducedhypergroup This is a hypergroup in which the equivalence class of any element with respect to adetermined fuzzy set is a singleton The most well known fuzzy set considered on a hypergroup is thegrade fuzzy set used for the study of the fuzzy grade of a hypergroup Based on this in the second part ofthe paper we study the fuzzy reducibility of some particular classes of crisp hypergroups with respect tothe grade fuzzy set

Keywords fuzzy set reducibility grade fuzzy set

1 Introduction

In the algebraic hypercompositional structures theory the most natural link with the classicalalgebraic structures theory is assured by certain equivalences that work as a bridge between boththeories More explicitly the quotient structure modulo the equivalence β defined on a hypergroupis always a group [1] the quotient structure modulo the equivalence Γ defined on a hyperring is a ring [2]while a commutative semigroup can be obtained factorizing a semihypergroup by the equivalence γ [3]A more completed list of such equivalences is very clearly presented in [4] All the equivalences having thisproperty ie they are the smallest equivalence relations defined on a hypercompositional structure suchthat the corresponding quotient (modulo that relation) is a classical structure with the same behaviourare called fundamental relations while the associated quotients are called fundamental structures Thestudy of the fundamental relations represents an important topic of research in HypercompositionalAlgebra also nowadays [5ndash7] But this is not the unique case when the name ldquofundamentalrdquo is given to anequivalence defined on a hyperstructure Indeed there exist three other equivalences called fundamentalby Jantosciak [8] who observed that unlike what happens in classical algebraic structures two elementscan play interchangeable roles with respect to a hyperoperation In other words the hyperoperation doesnot distinguish between the action of the given elements These particular roles have been translated bythe help of the following three equivalences We say that two elements x and y in a hypergroup (H ) are

1 operationally equivalent if their hyperproducts with all elements in H are the same2 inseparable if x belongs to the same hyperproducts as y belongs to3 essentially indistinguishable if they are both operationally equivalent and inseparable

75


Then a hypergroup is called reduced [8] if the equivalence class of any element with respect to theessentially indistinguishable relation is a singleton Besides the associated quotient structure (withrespect to the same fundamental relation) is a reduced hypergroup called the reduced form of theoriginal hypergroup Jantosciak explained the role of these fundamental relations by a very simpleand well known result Define on the set H = Ztimes Zlowast where Z is the set of integers and Zlowast = Z 0the equivalencesimthat assigns equivalent fractions in the same class (x y) sim (u v) if and only if xv = yufor (x y) (u v) isin H Endow H with a hypercompositional structure considering the hyperproduct(w x) (y z) = (wz + xy xz)sim where by xsim we mean the equivalence class of the element x with respectto the equivalence sim It follows that the equivalence class of the element (x y) isin H with respect to allthree fundamental relations defined above is equal to the equivalence class of (x y) with respect to theequivalence sim Therefore H is not a reduced hypergroup while its reduced form is isomorphic with Qthe set of rationals

Motivated by this example in the same article [8] Jantosciak proposed a method to obtain allhypergroups having a given reduced hypergroup as their reduced form This aspect of reducibility ofhypergroups has been later on considered by Cristea et al [9ndash11] obtaining necessary and sufficientcondition for a hypergroup associated with a binary or nminusary relation to be reduced The same authorstudied the regularity aspect of these fundamental relations and started the study of fuzzyfication of theconcept of reducibility [12] This study can be considered in two different directions corresponding tothe two different approaches of the theory of hypergroups associated with fuzzy sets An overview ofthis theory is covered by the monograph [13] written by Davvaz and Cristea the only one on this topicpublished till now The fuzzy aspect of reducibility could be investigated in two ways by studying theindistinguishability between the elements of a fuzzy hypergroup (ie a structure endowed with a fuzzyhyperoperation) or by studying the indistinguishability between the images of the elements of a crisphypergroup through a fuzzy set By a crisp hypergroup we mean a hypergroup but the attribute ldquocrisprdquo isused to emphasize that the structure is not fuzzy The study conducted in this article follows the seconddirection while the first direction will be developed in our future works

The aim of this note is to study the concept of fuzzy reduced hypergroup as a crisp hypergroup which isfuzzy reduced with respect to the associated fuzzy set One of the most known fuzzy sets associated with ahypergroup is the grade fuzzy set micro introduced by Corsini [14] It was studied by Corsini and Cristea [15]in order to define the fuzzy grade of a hypergroup as the length of the sequence of join spaces and fuzzy setsassociated with the given hypergroup For any element x in a hypergroup H the value micro (x) is defined asthe average value of the reciprocals of the sizes of all hyperproducts containing x The properties of thisparticular fuzzy set in particular those related to the fuzzy grade have been investigated for several classesof finite hypergroups as complete hypergroups non-complete 1-hypergroups or ips hypergroups (iehypergroups with partial scalar identities) Inspired by all these studies first we introduce the definitionof fuzzy reduced hypergroups and present some combinatorial aspects related to them Then we focus onthe fuzzy reducibility of ips hypergroups complete hypergroups and non-complete 1-hypergroups withrespect to the grade fuzzy set micro Theorem 3 states that any proper complete hypergroup is not reducedeither fuzzy reduced with respect to micro Regarding the ips hypergroups we show that they are reducedbut not fuzzy reduced with respect to micro (see Theorems 4 and 5) Finally we present a general methodto construct a non-complete 1-hypergroup that is not reduced either fuzzy reduced with respect to microSome conclusions and new research ideas concerning this study are gathered in the last section

2 Review of Reduced Hypergroups

In this section we briefly recall the basic definitions which will be used in the following as well asthe main properties of reduced hypergroups We start with the three fundamental relations defined by

76


Jantosciak [8] on an arbitrary hypergroup Throughout this note by a hypergroup (H ) we mean anon-empty set H endowed with a hyperoperation usually denoted as H times H rarr Plowast (H) satisfying theassociativity ie for all a b c isin H there is (a b) c = a (b c) and the reproduction axiom ie for alla isin H there is a H = H a = H where Plowast (H) denotes the set of all non-empty subsets of H

Definition 1 ([8]) Two elements x y in a hypergroup (H ) are called

1 operationally equivalent or by short o-equivalent and write x simo y if x a = y a and a x = a y for anya isin H

2 inseparable or by short i-equivalent and write x simi y if for all a b isin H x isin a blArrrArr y isin a b3 essentially indistinguishable or by short e-equivalent and write x sime y if they are operationally equivalent

and inseparable

Definition 2 ([8]) A reduced hypergroup has the equivalence class of any element with respect to the essentiallyindistinguishable relation sime a singleton ie for any x isin H there is xe = x

Example 1 Let (H ) be a hypergroup where the hyperoperation ldquordquo is defined by the following table

a b c da a a a b c a b db a a a b c a b dc a b c a b c a b c c dd a b d a b d c d a b d

One notices that a simo b because the lines (and columns) corresponding to a and b are exactly the same therebyao = bo = a b while co = c and do = d But on the other side the equivalence class of any element inH with respect to the relation simi is a singleton as well as with respect to the relation sime by consequence (H )is reduced

In [11] Cristea et al discussed about the regularity of these fundamental relations proving that ingeneral none of them is strongly regular This means that the corresponding quotients modulo theseequivalences are not classical structures but hypergroups Moreover Jantosciak [8] established thefollowing result

Theorem 1 ([8]) For any hypergroup H the associated quotient hypergroup H sime is a reduced hypergroupcalled the reduced form of H

Motivated by this property Jantosciak concluded that the study of the hypergroups can be dividedinto two parts the study of the reduced hypergroups and the study of all hypergroups having the samereduced form [8]

3 Fuzzy Reduced Hypergroups

As already mentioned in the introductory part of this article the extension of the concept of reducibilityto the fuzzy case can be performed on a crisp hypergroup endowed with a fuzzy set by defining similarlyto the classical case three equivalences as follows

Definition 3 Let (H ) be a crisp hypergroup endowed with a fuzzy set micro For two elements x y isin H we say that

1 x and y are fuzzy operationally equivalent and write x sim f o y if for any a isin H micro(x a) = micro(y a) andmicro(a x) = micro(a y)

77


2 x and y are fuzzy inseparable and write x sim f i y if micro(x) isin micro(a b)lArrrArr micro(y) isin micro(a b) for a b isin H3 x and y are fuzzy essentially indistinguishable and write x sim f e y if they are fuzzy operationally equivalent

and fuzzy inseparable

Definition 4 The crisp hypergroup (H ) is called fuzzy reduced if the equivalence class of any element in H withrespect to the fuzzy essentially indistinguishable relation is a singleton ie

for all x isin H x f e = x

Notice that the notion of fuzzy reducibility of a hypergroup is strictly connected with the definitionof the involved fuzzy set

Remark 1 It is easy to see that for each hypergroup H endowed with an arbitrary fuzzy set micro the followingimplication holds for any a b isin H

a simo brArr a sim f o b

Remark 2 (i) First of all let us better explain the meaning of micro (a b) for any a b isin H and any arbitrary fuzzyset micro defined on H Generally a b is a subset of H so micro (a b) is the direct image of this subset through the fuzzyset micro ie micro(a b) = micro (x) | x isin a b

From here two things need to be stressed on Firstly if a b is a singleton ie a b = c then micro (a b)is a set containing the real number micro (c) Therefore we can write micro (c) isin micro (a b) but it is not correct writingmicro(c) = micro(a b) because the first member is a real number while the second one is a set containing the realnumber micro (c)

Secondly if a isin x y then clearly micro (a) isin micro (x y) but not also viceversa because it could happen thatmicro (a) isin micro (xprime yprime) for a 6isin xprime yprime

(ii) Generally a simi b 6rArr a sim f i b as illustrated in the next example Indeed if a simi b then a isin x y ifand only if b isin x y But it can happen that micro (a) isin micro (xprime yprime) with a 6isin xprime yprime so also b 6isin xprime yprime And ifmicro(a) 6= micro(b) then micro (b) 6isin micro (xprime yprime) thus a 6simi b

(iii) Finally it is simple to see the implication micro (a) = micro (b)rArr a sim f i b

The following example illustrates all the issues in the above mentioned remark

Example 2 ([16]) Let (H ) be a hypergroup represented by the following commutative Cayley table

e a1 a2 a3e e a1 a2 a3 a2 a3a1 a2 a3 e ea2 a1 a1a3 a1

One notices immediately that a2 simi a3 while a1 6simi a2

(a) Define now on H the fuzzy set micro as follows micro (e) = 1 micro (a1) = micro (a2) = 03 micro (a3) = 05 Since micro (a1) =

micro (a2) it follows that a1 sim f i a2 Moreover since e a1 = a1 we have

micro (e a1) = micro (a1) = 03 3 micro (a2)

while it is clear that micro (a3) isin micro (e a1) so a2 6sim f i a3

78


(b) If we define on H the fuzzy set micro by taking micro (e) = micro (a1) = 1 micro (a2) = micro (a3) =13 it follows that e sim f i a1

and a2 sim f i a3

Once again it is evident that the three equivalences sim f o sim f i and sim f e are strictly related with thedefinition of the fuzzy set considered on the hypergroup

Now we will present an example of an infinite hypergroup and study its fuzzy reducibility

Example 3 Consider the partially ordered group (Z+le) with the usual addition and orderings of integersDefine on Z the hyperoperation a lowast b = x isin Z | a + b le x Then (Z lowast) is a hypergroup [17] Define now onZ the fuzzy set micro as follows micro(0) = 0 and micro(x) = 1

|x| for any x 6= 0 We obtain x f o = x for any x isin Ztherefore (Z lowast) is fuzzy reduced with respect to micro Indeed for two arbitrary elements x and y in Z we have x sim f o yif and only if micro(x lowast a) = micro(y lowast a) for any a isin Z where micro(x lowast a) = 1

|x+a| 1

|x+a+1| 1

|x+a+2| and similarly

micro(y lowast a) = 1|y+a|

1|y+a+1| Since a is an arbitrary integer for any x and y we always find a suitable integer a

such that x + a gt 0 and y + a gt 0 This means that the sets micro(x lowast a) and micro(y lowast a) contain descending sequences ofpositive integers so they are equal only when x = y Therefore x sim f o y

In the following we will study the fuzzy reducibility of some particular types of finite hypergroupswith respect to the grade fuzzy set micro defined by Corsini [14] We recall here its definition With any crisphypergroupoid (H ) (not necessarily a hypergroup) we may associate the fuzzy set micro considering for anyu isin H

micro(u) =

sum(xy)isinQ(u)

1|x y|

q(u) (1)

where Q(u) = (a b) isin H2 | u isin a b and q(u) = |Q(u)| By convention we take micro(u) = 0anytime when Q(u) = empty In other words the value micro(u) is the average value of reciprocals of the sizesof all hyperproducts x y containing the element u in H In addition sometimes when we will refer toformula (1) we will denote its numerator by A(u) while the denominator is already denoted by q(u)

Remark 3 As already explained in Remark 2 (ii) generally for an arbitrary fuzzy set x simi y 6rArr x sim f i y while theimplication holds if we consider the grade fuzzy set micro Indeed if x simi y then x isin a b if and only if y isin a band therefore Q(x) = Q(y) implying that q(x) = q(y) and moreover A(x) = A(y) This leads to the equalitymicro(x) = micro(y) By consequence based on Remark 2 (iii) it holds x sim f i y with respect to micro

Example 4 Let us consider now a total finite hypergroup H ie x y = H for all x y isin H It is easy to see thatx sime y for any x y isin H meaning that xe = H for any x isin H Thus a total hypergroup is not reduced What canwe say about the fuzzy reducibility with respect to the grade fuzzy set micro

For any u isin H there is

micro(u) =|H|2 1

|H||H|2

=1|H|

Since x sim y for any x y isin H it follows that x sim f o y for any x y isin H Then it is clear that micro(x) = micro(y)for all x y isin H implying that x sim f i y for all x y isin H Concluding it follows that any total finite hypergroup isneither reduced nor fuzzy reduced

Based now on Remarks 1 and 3 the following assertion is clear

79


Corollary 1 If (H ) is a not reduced hypergoup then it is also not fuzzy reduced with respect to the grade fuzzyset micro

31 Fuzzy Reducibility in Complete Hypergroups

The complete hypergroups form a particular class of hypergroups strictly related with the joinspaces and the regular hypergroups Their definition is based on the notion of complete part introducedin Koskas [1] with the main role to characterize the equivalence class of an element under the relationβlowast More exactly a nonempty set A of a semihypergroup (H ) is called a complete part of H if for anynatural number n and any elements a1 a2 an in H the following implication holds

A capn

prodi=1

ai 6= emptyrArrn

prodi=1

ai sube A

We may say as it was mentioned in the review written by Antampoufis et al [18] that a complete partA absorbs all hyperproducts of the elements of H having non-empty intersection with A The intersectionof all complete parts of H containing the subset A is called the complete closure of A in H and denotedby C (A)

Definition 5 A hypergroup (H ) is called a complete hypergroup if for any x y isin H there is C (x y) = x y

As already explained in the fundamental book on hypergroups theory [19] and the othermanuscripts related with complete hypergroups [1620] in practice it is more useful to use the followingcharacterization of the complete hypergroups

Theorem 2 ([19]) Any complete hypergroup may be constructed as the union H =⋃

gisinGAg of its subsets where

(1) (G middot) is a group(2)The family Ag |g isin G is a partition of G ie for any (g1 g2) isin G2 g1 6= g2 there is Ag1 cap Ag2 = empty(3) If (a b) isin Ag1 times Ag2 then a b = Ag1g2

Example 5 The hypergroup presented in Example 2 is complete where the group G = (Z3+) and the partitionset contains A0 = e A1 = a1 and A2 = a2 a3 It is clear that all conditions in Theorem 2 are fulfilled

Let (H ) be a proper complete hypergroup (ie H is not a group) Define now on H the equivalenceldquosimrdquo by

x sim ylArrrArr existg isin G such that x y isin Ag (2)

Proposition 1 On a proper complete hypergroup (H ) the equivalencesimin (2) is a representation of theessentially indistinguishability equivalence sime

Proof By Theorem 2 one notices that for any element x isin H there exists a unique g isin G such that x isin Agx In the following we will denote this element by gx First suppose that x sim y ie there exists gx = gy isin Gsuch that x y isin Agx For any arbitrary element a isin H we can say that a isin Aga with ga isin G and by thedefinition of the hyperproduct in the complete hypergroup (H ) there is x a = Agx ga = Agyga = y a(and similarly a x = a y) implying that x simo y (ie x and y are operationally equivalent) Secondlyfor any x isin a b = Agagb cap Agx it follows that gagb = gx but gx = gy so y isin Agagb = a b Thereby

80


x isin a b if and only if y isin a b meaning that x simi y (ie x and y are inseparable) We have proved thatsimsubesime

Conversely let us suppose that x sime y Since x and y are inseparable ie x isin a b if and onlyif y isin a b we may write x y isin Agagb Therefore there exists gx = ga middot gb isin G such that x y isin Agx so x sim y

Example 6 If we continue with Example 2 we notice that the equivalence classes of the elements of H withrespect to the equivalencesimdefined in (2) are e = e a1 = a1 a2 = a2 a3 = a3 Regarding now theessentially indistinguishability equivalence sime we have the same equivalences classes ee = e a1e = a1a2e = a2 a3 = a3e This is clear from the Cayley table of the hypergroup the lines corresponding to a2 and a3 arethe same and every time a2 and a3 are in the same hyperproduct so they are equivalent Since these properties arenot satisfied for a1 and e their equivalence classes are singletons

Proposition 2 Let (H ) be a proper complete hypergroup and consider on H the grade fuzzy set micro Then simsubsim f e(with respect to the fuzzy set micro)

Proof By the definition of the grade fuzzy set micro one obtains that

micro(x) =1|Agx |

for any x isin H (3)

Take now x y isin H such that x sim y There exists gx isin G such that x y isin Agx thereby micro (x) = micro (y)and by Remark 2 (ii) we have x sim f i y Moreover by Proposition 1 there is x simo y and by Remark 1 we getthat x sim f o y Concluding we have proved that x sim yrArr x sim f e y with respect to micro

Theorem 3 Any proper complete hypergroup is not reduced either fuzzy reduced with respect to the grade fuzzy set

Proof Since (H ) is a proper complete hypergroup there exists at least one element g in G such that|Ag| ge 2 ie there exist two distinct elements a and b in H with the property that a sim b Then a sime b anda sim f e b meaning that (H ) is not reduced either fuzzy reduced with respect to the grade fuzzy set micro

32 Fuzzy Reducibility in ips Hypergroups

An ips hypergroup is a canonical hypergroup with partial scalar identities The name given byCorsini [21] originally comes from Italian the abbreviation ldquoipsrdquo standing for ldquoidentitagrave parziale scalarerdquo(in English partial scalar identity) First we dwell on the terminology connected with the notion of identityin a hypergroup (H ) An element x isin H is called a scalar if |x y| = |y x| = 1 for any y isin HAn element e isin H is called partial identity of H if it is a left identity (ie there exists x isin H such thatx isin e x) or a right identity (ie there exists y isin H such that y isin y e) We denote by Ip the set of all partialidentities of H Besides for a given element x isin H a partial identity of x is an element u isin H such thatx isin x u cup u x The element u isin H is a partial scalar identity of x whenever from x isin x u it followsx = x u and from x isin u x it follows that x = u x We denote by Ip (x) the set of all partial identities ofx by Ips (x) the set of all partial scalar identities of x and by Sc(H) the set of all scalars of H It is obviousthat Ips(x) = Ip(x) cap Sc(H)

Remark 4 The term ldquopartialrdquo here must not be confused with the ldquoleft or rightrdquo (identity) but it must be connectedwith the fact that the element u has a partial behaviour of identity with respect to the element x So u is not aleftright (ie partial) identity for the hypergroup H Besides an ips hypergroup is a commutative hypergroupso the concept of partial intended as leftright element satisfying a property (ie leftright unit) has no sense It is

81


probably better to understand an element u having the property of being partial identity for x as an element having asimilar behaviour as an identity but only with respect to x so a partial role of being identity

Let us recall now the definition of an ips hypergroup All finite ips hypergroups of order less than9 have been determined by Corsini [21ndash23]

Definition 6 A hypergroup (H ) is called ips hypergroup if it satisfies the following conditions

1 It is canonical ie

bull it is commutativebull it has a scalar identity 0 such that 0 x = x for any x isin Hbull every element x isin H has a unique inverse xminus1 isin H that is 0 isin x xminus1bull it is reversible so y isin a x =rArr x isin aminus1 y for any a x y isin H

2 It satisfies the relation for any a x isin H if x isin a x then a x = x

The most useful properties of ips hypergroups are gathered in the following result

Proposition 3 ([21]) Let (H ) be an ips hypergroup

1 For any element x in H the set x xminus1 is a subhypergroup of H2 For any element x in H different from 0 we have or x is a scalar of H or there exists u isin Sc(H) 0

such that u isin x xminus1 Moreover |Sc(H)| ge 23 If x is a scalar of H then the set of all partial scalar identities of x contains just 0

If x is not a scalar of H then Ips(x) sub Sc(H) cap x xminus1 and therefore |Ips(x)| ge 2

Proposition 4 Let (H ) be an ips hypergroup For any scalar u isin H and for any element x isin H there exists aunique y isin H such that u isin x y

Proof The existence part immediately follows from the reproducibility property of H For proving theunicity assume that there exist y1 y2 isin H y1 6= y2 such that u isin x y1 cap x y2 Then by reversibilityit follows that y1 y2 isin xminus1 u Since u is a scalar element we get |xminus1 u| = 1 and then y1 = y2 =

xminus1 u

Example 7 ([21]) Let us consider the following ips hypergroup (H )

H 0 1 2 30 0 1 2 31 1 2 0 3 12 2 0 3 1 23 3 1 2 0

Here we can notice that 0 is the only one identity of H In addition Sc(H) = 0 3 and 0 isin 0 u only foru = 0 so Ips(0) = 0 Also only for u = 0 there is 3 isin 3 u thus Ips(3) = 0 (In general if x isin Sc(H)

then Ips(x) = 0 according with Proposition 3) Similarly one gets Ip(1) = Ip(2) = 0 3 and sinceSc(H) = 0 3 it follows that Ips(1) = Ips(2) = 0 3

82


Note that in an ips hypergroup the Jantosciak fundamental relations have a particular meaningin the sense that for any two elements there is

a simo blArrrArr a simi blArrrArr a sime blArrrArr a = b

By consequence one obtains the following result

Theorem 4 Any ips hypergroup is reduced

In the following we will discuss the fuzzy reducibility of an ips hypergroup with respect to thegrade fuzzy set micro

Theorem 5 Any ips hypergroup is not fuzzy reduced with respect to the fuzzy set micro

Proof Since any ips hypergroup contains at least one non-zero scalar take arbitrary such a u isin Sc(H)We will prove that u sim f i 0 and u sim f o 0 therefore |0 f e| ge 2 meaning that H is not fuzzy reduced

First we will prove that for any u isin Sc(H) there is micro(0) = micro(u) equivalently with u sim f i 0 For doing

this based on the fact that micro(x) = A(x)q(x) for all x isin H we show that A(0) = A(u) and q(0) = q(u)

Let us start with the computation of q(0) and q(u) If 0 isin x y it follows that y isin xminus1 0 that is y =

xminus1 and then Q(0) = (x y) isin H2 | 0 isin x y = (x xminus1) | x isin H Thereby q(0) = |Q(0)| = n = |H|On the other hand by Proposition 4 we have q(u) = n = |H| (since for any x isin H there exists a uniquey isin H such that u isin x y )

Let us calculate now A(0) By formula (1) we get that

A(0) = sum(xy)isinQ(0)

1|x y| = sum

xisinH

1|x xminus1| =

sumaisinSc(H)

1|a aminus1| + sum

x 6isinSc(H)

1|x xminus1| = |Sc(H)|+ sum

x 6isinSc(H)

1|x xminus1|

Since for u isin Sc(H) existx 6isin Sc(H) such that u isin x xminus1 cap Ips(x) we similarly get that

A(u) = |Sc(H)|+ sumx 6isinSc(H)

1|x xminus1|

and it is clear that A(0) = A(u) so micro(0) = micro(u) Therefore 0 sim f i uIt remains to prove the second part of the theorem that is 0 sim f o u equivalently with micro(0 x) =

micro(u x) forallx isin HIf x isin Sc(H) then u x isin Sc(H) and by the first part of the theorem there is micro(u x) = micro(0) =

micro(x) = micro(0 x)If x 6isin Sc(H) then since Sc(H) sub Ips(a) for any a 6isin Sc(H) it follows that Sc(H) sub Ips(x) so u x =

x and then micro(u x) = micro(x) = micro(0 x) Now the proof is complete

33 Fuzzy Reducibility in Non-Complete 1-Hypergroups

In this subsection we will study the reducibility and fuzzy reducibility of some particular finitenon-complete 1-hypergroups defined and investigated by Corsini Cristea [24] with respect to their fuzzygrade A hypergroup H is called 1-hypergroup if the cardinality of its heart ωH is 1

83


The general construction of this particular hypergroup is the following one Consider the set H =

Hn = ecup Acup B where A = a1 aα and B = b1 bβ with α β ge 2 and n = α+ β+ 1 such thatA cap B = empty and e isin A cup B Define on H the hyperoperation ldquordquo by the following rule

bull for all a isin A a a = b1bull for all (a1 a2) isin A2 such that a1 6= a2 set a1 a2 = Bbull for all (a b) isin Atimes B set a b = b a = ebull for all (b bprime) isin B2 there is b bprime = Abull for all a isin A set a e = e a = Abull for all b isin B b e = e b = B andbull e e = e

Hn is an 1-hypergroup which is not completeWe will discuss the (fuzzy) reducibility of this hypergroup for different cardinalities of the sets A

and B(1) Let us suppose n = |H6| = 6 where H = H6 = e cup A cup B α = |A| = 2 β = |B| = 3 A cap B = empty

e isin A cup B with A = a1 a2 B = b1 b2 b3 Thus the Cayley table of (H6 ) is the following one

H e a1 a2 b1 b2 b3e e A A B B Ba1 b1 B e e ea2 b1 e e eb1 A A Ab2 A Ab3 A

From the table we notice immediately that the elements b2 and b3 are essentially indistinguishablewhile the equivalence class with respect to the sime relation of all other elements is a singleton Thereby H isnot reduced

Calculating now the values of the grade fuzzy set micro one obtains micro (e) = 1 micro (a1) = micro (a2) =

05 micro (b1) = 0467 micro (b2) = micro (b3) = 0333 Since micro(a1) = micro(a2) it follows that a1 sim f i a2 But micro(a1 a1) =

micro(b1) = micro(b1) while micro(a1 a2) = micro(B) = micro(b1) micro(b2) so micro(a1 a1) 6= micro(a1 a2) meaning thata1 6sim f o a2 that is a1 6sim f e a2

On the other side we have b2 sim f o b3 because they are also operationally equivalent and b2 sim f i b3

because micro(b2) = micro(b3) This is equivalent with b2 sim f e b3 and therefore H is not reduced with respect to micro(2) Consider now the most general case The Cayley table of the hypergroup H is the following one

H e a1 a2 middot middot middot aα b1 b2 middot middot middot bβ

e e A A middot middot middot A B B middot middot middot Ba1 b1 B middot middot middot B e e middot middot middot ea2 b1 middot middot middot B e e middot middot middot e

aα b1 e e middot middot middot eb1 A A middot middot middot Ab2 A middot middot middot A

bβ A

84


As already calculated in [24] there is micro(e) = 1 micro (ai) = 1α for any i = 1 α while micro (b1) =

α2 + αβ + 2βminus α

β(α2 + 2β) and micro

(bj)= 1

β for any j = 2 β As in the previous case we see that any two

elements in B b1 are operational equivalent by consequence also fuzzy operational equivalentand indistinguishable and fuzzy indistinguishable (because their values under the grade fuzzy set micro

are the same) Concluding this non complete 1-hypergroup is always not reduced either not fuzzyreduced with respect to micro


The pioneering paper of Rosenfeld [25] on fuzzy subgroups has opened a new perspective of thestudy of algebraic structures (hyperstructures) using the combinatorial properties of fuzzy sets Regardingthis there are two distinct views one concerns the crisp (hyper)structures endowed with fuzzy setsand the other one is related to fuzzy (hyper)structures that are sets on which fuzzy (hyper)operationsare defined Several classical algebraic concepts have then been extended in the framework of these twolines of research reducibility in hypergroups defined by Jantosciak [8] already in 1990 being one of themIn this paper we have focussed on the study of fuzzy reducibility in hypergroups extending the concept ofreducibility to hypergroups endowed with fuzzy sets For a better understanding of this subject we havechosen the grade fuzzy set micro [14] and have investigated the fuzzy reducibility of some particular classesof hypergroups those with scalar partial identities complete hypergroups and special non-complete1-hypergroups Several connections between reducibility and fuzzy reducibility (with respect to micro) havebeen underlined and motivated by several examples

In our future work following the other connection between fuzzy sets and (hyper)structures we aimto describe the reducibility of fuzzy hypergroups [26] by studying the reduced fuzzy hypergroups they arefuzzy hypergroups which are reduced Moreover these aspects will also be investigated in the case of themimic fuzzy hypergroups [27]

Author Contributions Conceptualization MK and IC methodology MK and IC investigation MK and ICwritingmdashoriginal draft preparation MK writingmdashreview and editing IC supervision IC funding acquisition ICAll authors have read and agreed to the published version of the manuscript

Funding The second author acknowledges the financial support from the Slovenian Research Agency (research corefunding No P1-0285)


References

1 Koskas M Groupoides demi-hypergroupes et hypergroupes J Math Pure Appl 1970 49 155ndash1922 Vougiouklis T The Fundamental Relation in Hyperrings the General Hyperfield In Algebraic Hyperstructures

and Applications (Xanthi 1990) World Sci Publishing Teaneck NJ USA 1991 pp 203ndash2113 Freni D A new characterization of the derived hypergroup via strongly regular equivalences Commun Algebra

2002 30 3977ndash3989 [CrossRef]4 Norouzi M Cristea I Fundamental relation on m-idempotent hyperrings J Open Math 2017 15 1558ndash1567

[CrossRef]5 De Salvo M Fasino D Freni D Lo Faro G On hypergroups with a β-class of finite height Symmetry 2020

12 168 [CrossRef]6 Zadeh A Norouzi M Cristea I The commutative quotient structure of m-idempotent hyperrings An Stiintifice

Univ Ovidius Constanta Ser Mat 2020 28 219ndash2367 Ameri R Eyvazi M Hoskova-Mayerova S Superring of Polynomials over a Hyperring Mathematics 2019

7 902 [CrossRef]

85


8 Jantosciak J Reduced hypergroups In Proceedings of the Algebraic Hyperstructures and ApplicationsProceedings of 4th International Congress Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991pp 119ndash122

9 Cristea I Several aspects on the hypergroups associated with n-ary relations An Stiintifice Univ OvidiusConstanta Ser Mat 2009 17 99ndash110

10 Cristea I Stefanescu M Hypergroups and n-ary relations Eur J Comb 2010 31 780ndash789 [CrossRef]11 Cristea I Stefanescu M Angheluta C About the fundamental relations defined on the hypergroupoids

associated with binary relations Eur J Comb 2011 32 72ndash81 [CrossRef]12 Cristea I Reducibility in hypergroups (crisp and fuzzy case) Int J Algebr Hyperstructures Its Appl 2015

2 67ndash7613 Davvaz B Cristea I Fuzzy Algebraic Hyperstructures-An Introduction In Studies in Fuzziness and Soft

Computing Springer Cham Switzerland 2015 Volume 32114 Corsini P A new connection between hypergroups and fuzzy sets Southeast Asian Bull Math 2003 27 221ndash22915 Corsini P Cristea I Fuzzy grade of ips hypergroups of order 7 Iran J Fuzzy Syst 2004 1 15ndash3216 Cristea I Complete hypergroups 1-hypergroups and fuzzy sets An Stiintifice Univ Ovidius Constanta Ser Mat

2002 10 25ndash3717 Novak M Cristea I Cyclicity in EL-hypergroups Symmetry 2018 10 611 [CrossRef]18 Antampoufis N HoškovaacutendashMayerovaacute Š A brief survey on the two different approaches of fundamental

equivalence relations on hyperstructures Ratio Math 2017 33 47ndash6019 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Tricesimo Italy 199320 Davvaz B Hassani Sadrabadi E Cristea I Atanassovrsquos intuitionistic fuzzy grade of the complete hypergroups

of order less than or equal to 6 Hacet J Math Stat 2015 44 295ndash31521 Corsini P Sugli ipergruppi canonici finiti con identitagrave parziali scalari Rend Circ Mat Palermo 1987 36 205ndash219

[CrossRef]22 Corsini P (ips) Ipergruppi di ordine 6 Ann Sc De LrsquoUniv Blaise Pascal Clermont-II 1987 24 81ndash10423 Corsini P (ips) Ipergruppi di ordine 7 Atti Sem Mat Fis Univ Modena 1986 34 199ndash21624 Corsini P Cristea I Fuzzy sets and non complete 1-hypergroups An Stiintifice Univ Ovidius Constanta Ser Mat

2005 13 27ndash5425 Rosenfeld A Fuzzy subgroups J Math Anal Appl 1971 35 512ndash517 [CrossRef]26 Sen MK Ameri R Chowdhury G Fuzzy hypersemigroups Soft Comput 2008 12 891ndash900 [CrossRef]27 Massouros CG Massouros GG On 2-element fuzzy and mimic fuzzy hypergroups J AIP Conf Proc 2012

1479 2213ndash2216

copy 2020 by the authors Licensee MDPI Basel Switzerland This article is an open access articledistributed under the terms and conditions of the Creative Commons Attribution (CC BY)license (httpcreativecommonsorglicensesby40)

86

mathematics

Article

Regular Parameter Elements and Regular Local Hyperrings

Hashem Bordbar and Irina Cristea

Citation Bordbar H Cristea I

Regular Parameter Elements and

Regular Local Hyperrings



Received 31 December 2020

Accepted 25 January 2021

Published 26 January 2021

Publisherrsquos Note MDPI stays neu-

tral with regard to jurisdictional clai-

ms in published maps and institutio-

nal affiliations

Copyright copy 2021 by the authors Li-

censee MDPI Basel Switzerland


distributed under the terms and con-

ditions of the Creative Commons At-

tribution (CC BY) license (https


40)

Centre for Information Technologies and Applied Mathematics University of Nova Gorica5000 Nova Gorica Slovenia hashembordbarungsi Correspondence irinacristeaungsi or irinacriyahoocouk Tel +386-0533-15-395

Abstract Inspired by the concept of regular local rings in classical algebra in this article we ini-tiate the study of the regular parameter elements in a commutative local Noetherian hyperringThese elements provide a deep connection between the dimension of the hyperring and its primaryhyperideals Then our study focusses on the concept of regular local hyperring R with maximalhyperideal M having the property that the dimension of R is equal to the dimension of the vectorialhyperspace M

M2 over the hyperfield RM Finally using the regular local hyperrings we determine the

dimension of the hyperrings of fractions

Keywords Krasner hyperring primemaximal hyperideal length of hypermodules regular localhyperring parameter elements

1 Introduction

A regular local ring is known as a Noetherian ring with just one maximal ideal gen-erated by n elements where n is the Krull dimension of the ring This is equivalent withthe condition that the ring R has a system of parameters that generate the maximal idealcalled regular system of parameters or regular parameter elements [1] In commutativealgebra the theory of regular local rings plays a fundamental role and it has been de-veloped since the late 30s and early 40s of the last century thanks to the studies of twogreat mathematicians Wolfgang Krull and Oscar Zariski This happened just some yearsafter that Frederic Marty introduced hypergroups as a generalization of groups in sucha way that the single-valued group operation was extended to a hyperoperation ie toa multi-valued operation It is important to stress the fact that not all the properties ofthe group such as the existence of the neutral element and the inverse have been exactlytransferred to the hypergroups meaning that it is not obligatory that a hypergroup containsa neutral element or inverses These requirements were requested later on for so calledcanonical hypergroups that were defined as the additive parts of the Krasner hyperringsand hyperfields [23]

The hyperring is a hypercompositional structure endowed with one hyperoperationnamely the addition and one binary operation namely the multiplication satisfyingcertain properties Krasner [2] introduced the concept of hyperring for the first time in1957 and investigated its applicability to the theory of valued fields There are someother types of hyperrings if the multiplication is a hyperoperation and the addition isa binary operation we talk about the multiplicative hyperrings defined by Rota [4] Ifboth the addition and the multiplication are hyperoperations with the additive part beinga canonical hypergroup then we have superrings [5] which were introduced by Mittasin 1973 [6] Until now the most well known and studied type of hyperrings is the Kras-ner hyperring that has a plentitude of applications in algebraic geometry [78] tropicalgeometry [9] theory of matroids [10] schemes theory [11] algebraic hypercurves [1213]hypermomographies [14] In addition the theory of hypermodules was extensively inves-tigated by Massouros [15] In this article the free and cyclic hypermodules are studiedand several examples are provided such as the one obtained as a quotient of a P-module

87


over a unitary ring P Recently Bordbar and his collaborators in [16ndash18] have introducedthe length and the support of hypermodules and studied some properties of them thatare used also in this paper Moreover the connection between the hyperringshyperfieldstheory and geometry is very clearly explained by Massouros in [19] This paper is a pylonin the current literature on algebraic hypercompositional algebra because it describes thedevelopment of this theory (with a lot of examples and explanation of the terminology)from the first definition of hypergroup proposed by F Marty to the hypergroups endowedwith more axioms and used now a days

In this paper the theory of regular local rings is applied in the context of commutativeNoetherian Krasner hyperrings with only one maximal hyperideal namely the localhyperrings We first define the regular parameter elements in an arbitrary commutativelocal hyperring of finite dimension These elements come up from the nice and deeprelation existing between the dimension of a local hyperring with maximal hyperideal Mand the set of the generators of its M-primary hyperideals (see Theorem 4) More preciselysince in a local hyperring R with maximal hyperideal M the dimension of the ring R isequal with the height of the hyperideal M ie dimR = htR M as stated in Corolarry 1 wecan say that the regular parameter elements are a consequence of the investigation of theheight of M and the set of the generators for M-primary hyperideals in R The other mainobjective of Section 3 is expressed by the result regarding the dimension of the quotienthyperrings (see Proposition 5) In Section 4 using the local hyeprring R with maximalhyperideal M and the structure of the quotient hyperring we introduce the hypermoduleMM2 over the hyperrings R and R

M respectively Since RM is a hyperfield we conclude that

MM2 is a vectorial hyperspace over the hyperfield R

M as a direct consequence of Theorem 6Moreover the investigation on the relation between the dimension of the hyperring R(equivalently the height htR M of the maximal hyperideal M) and the dimension of thevectorial hyperspace M

M2 conducts us to the definition of the regular local hyperrings Theyare exactly local Noetherian hyperrings with the property that the maximal hyperideal Mcan be generated by d elements where d is the dimension of the hyperring This followsfrom the main result of this section ie Theorem 7 saying that the dimension of a localNoetherian hyperring R is the smallest number of elements that generate an M-primaryhyperideal of R Finally we apply these results in the class of the hyperrings of fractionsie hyperrings of the form RP = Sminus1R where S = R P with P a prime hyperideal of S isa multiplicatively closed subset of R We prove that the height of the prime hyperideal P isequal to the height of the hyperideal Sminus1P in the hyperring of fractions SP (see Theorem 8)Final conclusions and some future works on this topic are gathered in the last section ofthe paper

2 Preliminaries

In this section we collect some fundamental results regarding hyperrings but for moredetails we refer the readers to [16ndash18] Throughout the paper R denotes a Krasner hyperringunless stated otherwise and we call it by short a hyperring It was introduced by Krasner [2]as follows

Definition 1 A (Krasner) hyperring is a hyperstructure (R+ middot) where

1 (R+) is a canonical hypergroup ie

(a) (a b isin RrArr a + b sube R)(b) (foralla b c isin R) (a + (b + c) = (a + b) + c)(c) (foralla b isin R) (a + b = b + a)(d) (exist0 isin R)(foralla isin R) (a + 0 = a)(e) (foralla isin R)(exist minus a isin R) (0 isin a + x hArr x = minusa)(f) (foralla b c isin R)(c isin a + brArr a isin c + (minusb))

2 (R middot) is a semigroup with a bilaterally absorbing element 0 ie

(a) (a b isin RrArr a middot b isin R)

88


(b) (foralla b c isin R) (a middot (b middot c) = (a middot b) middot c)(c) (foralla isin R) (0 middot a = a middot 0 = 0)

3 The product distributes from both sides over the sum

(a) (foralla b c isin R) (a middot (b + c) = a middot b + a middot c and (b + c) middot a = b middot a + c middot a)Moreover if (R middot) is commutative ie

4 (foralla b isin R) (a middot b = b middot a)the hyperring is called commutative

Definition 2 If every nonzero element in a hyperring R with multiplicative identity 1 is invert-ible ie

(i) (foralla isin R)(existaminus1 isin R) (a middot aminus1 = 1R)

then R is called a hyper f ield

Definition 3 A nonempty set I of a hyperring R is called a hyperideal if for all a b isin I andr isin R we have aminus b sube I and a middot r isin I A proper hyperideal M of a hyperring R is called a maximalhyperideal of R if the only hyperideals of R that contain M are M itself and R A hyperideal P ofa hyperring R is called a prime hyperideal of R if for every pair of elements a and b of R the factthat ab isin P implies either a isin P or b isin P In addition a nonzero hyperring R having exactly onemaximal hyperideal is called a local hyperring

Definition 4 A hyperring homomorphism is a mapping f from a hyperring R1 to a hyperring R2with units elements 1R1 and 1R2 respectively such that

1 (foralla b isin R) ( f (a +R1 b) = f (a) +R2 f (b))2 (foralla b isin R) ( f (a middotR1 b) = f (a) middotR2 f (b))3 f (1R1) = 1R2

Definition 5 Let f R rarr S be a hyperring homomorphism I be a hyperideal of R and J be ahyperideal of S

(i) The hyperideal lt f (I) gt of S generated by the set f (I) is called the extension of I and it isdenoted by Ie

(ii) The hyperideal fminus1(J) = a isin R | f (a) isin J is called the contraction of J and it is denotedby Jc It is known that if J is a prime hyperideal in S then Jc is a prime hyperideal in R

Definition 6 A prime hyperideal P of R is called a minimal prime hyperideal over a hyperideal Iof R if it is minimal (with respect to inclusion) among all prime hyperideals of R containing I Aprime hyperideal P is called a minimal prime hyperideal if it is a minimal prime hyperideal over thezero hyperideal of R

Definition 7 A hyperring R is called Noetherian if it satisfies the ascending chain conditionon hyperideals of R for every ascending chain of hyperideals I1 sube I2 sube I3 sube there existsN isin N such that In = IN for every natural number n ge N (this is equivalent to saying thatevery ascending chain of hyperideals has a maximal element) A hyperring R is called Artinianif it satisfies the descending chain condition on hyperideals of R for every descending chain ofhyperideals I1 supe I2 supe I3 supe there exists N isin N such that In = IN for every naturalnumber n ge N (this is equivalent to saying that every descending chain of hyperideals has aminimal element)

Definition 8 Let R be a hyperring with unit element 1 An R-hypermodule M is a commutativehypergroup (M+) together with a map RtimesM minusrarr M defined by

(a m) 7rarr a middotm = am isin M (1)

such that for all a b isin R and m1 m2 isin M we have

89


1 (a + b)m1 = am1 + bm12 a(m1 + m2) = am1 + am23 (ab)m1 = a(bm1)4 a0M = 0Rm1 = 0M5 1m1 = m1 where 1 is the multiplicative identity in R

Moreover if R is a hyperfield then M is called a vectorial hyperspace [20]

The next few results concern the concept of the radical of a hyperideal [16]

Definition 9 The radical of a hyperideal I of a hyperring R denoted by r(I) is defined as

r(I) = x | xn isin I for some n isin N

It can be proved that the radical of I is the intersection of all prime hyperideals of R containing IIn addition a hyperideal P in a hyperring R is called primary if P 6= R and the fact that xy isin Pimplies either x isin P or y isin r(P)

Lemma 1 Let I be a hyperideal of the hyperring R where r(I) is a maximal hyperideal of R ThenI is a primary hyperideal of R

Proposition 1 Let R be a Noetherian hyperring M a maximal hyperideal of R and let P be ahyperideal of R Then the following statements are equivalent

1 P is primary2 r(P) = M3 Mn sube P sube M for some n isin N

Definition 10 Let R be a commutative hyperring An expression of the type

P0 sub P1 sub sub Pn (2)

(note the strict inclusions) where P0 Pn are prime hyperideals of R is called a chain of primehyperideals of R the length of such a chain is the number of the ldquolinks between the terms of thechain that is 1 less than the number of prime hyperideals in the sequence

Thus the chain in (2) has length n Note that for a prime hyperideal P we considerP to be a chain with just one prime hyperideal of R of length 0 Since R is non-trivial itcontains at least one prime hyperideal so there certainly exists at least one chain of primehyperideals of R of length 0

Definition 11 The supremum of the lengths of all chains of prime hyperideals of R is called thedimension of R denoted by dimR

Definition 12 Let P be a prime hyperideal of a commutative hyperring R The height of P denotedby htRP is defined to be the supremum of the lengths of all chains

P0 sub P1 sub sub Pn

of prime hyperideals of R for which Pn = P if this supremum exists and it is infin otherwise

We conclude this preliminary section recalling the notion of vectorial hyperspace

Definition 13 ([20]) Let V be a vectorial hyperspace over a hyperfield F A linear combinationof vectors v1 v2 vn is a set of the form sumn

i=1 rivi Moreover for a vectorial hyperspace Vover a hyperfield F and A sube V A is called linearly independent if for every finite set of vectors

90


v1 v2 vn sube A 0 isin sumni=1 rivi implies ri = 0 for 1 le i le n If A is not linearly independent

then A is called linearly dependent

Definition 14 ([20]) Let V be a vectorial hyperspace over F The set A sube V is called a spanningset if each vector of V is contained in a linear combination of vectors from A We say V is thespan of A and write V = Span[A] In addition a basis for a vectorial hyperspace V is a subset Bof V such that B is both spanning and linearly independent set A vectorial hyperspace is finitedimensional if it has a finite basis Moreover the number of the elements in an arbitrary basis of avectorial hyperspace is called the dimension of the vectorial hyperspace denoted here by vdimV

Note that we denote the dimension of the vectorial hyperspace by vdimV (where ldquovrdquostays for ldquovectorialrdquo) in order to not confuse it with the dimension of a hyperring denotedhere as dimR

3 Regular Parameter Elements in Local Hyperrings

In this section we introduce the notion of the regular parameter elements in a localhyperring and by using representative examples we present some of their propertiesconnected with the length and the dimension of the related vectorial hyperspace

Definition 15 A unitary hyperring R is called principal hyperideal hyperdomain when it has nozero divisors and all its hyperideals are generated by a single element

M Krasner had the great idea to construct hyperrings and hyperfields as quotients ofrings and fields respectively called by him quotient hyperrings and quotient hyperfields whilethis construction is known in literature as Krasnerrsquos construction [3] The importance of thesehyperrings and hyperfields in Krasnerrsquos studies is very clear explained by G Massourosand Ch Massouros in [19] as well as their different names given by some authors [7ndash11]with the risk of creating confusions Therefore in order to keep the original terminologywe recommend to read the papers of Nakasis [21] and Massouros [152223]

Example 1 ([321]) Let (R+ middot) be a ring and G a normal subset of R (which means that rG = Grfor every r isin R) such that (G middot) is a group and the unit element of G is a unit element of R Definean equivalence relation sim= on R as follows r sim= s if and only if rG = sG Then the equivalenceclass represented by r is P(r) = s isin R | sG = rG = rG Define now a hyperoperation oplus on theset of all equivalence classes RG as follows P(r)oplus P(s) = P(t) | P(t) cap (P(r) + P(s)) 6= empty= tG | existg1 g2 isin G such that t = rg1 + sg2 = tG | tG sube rG + sG and define a binaryoperation on RG as rG middot sG = rsG(P(r) middot P(s) = P(rs)) Then (RGoplus middot) forms a hyperringMoreover if we choose R to be a field then we get that (RGoplus middot) is a hyperfield

Notice that the condition on the normality of G can be substituted with a more general one ierGsG = rsG for every r s isin Rlowast which is practically equivalent to the normality of G only whenthe multiplicative semigroup is a group so only when R is a field as Massouros proved in [22]

Proposition 2 Let R be a principal hyperideal hyperdomain and let p isin R minus 0 Then thefollowing statements are equivalent

(i) pR is a maximal hyperideal of R(ii) pR is a non-zero prime hyperideal of R

Proof (i) minusrarr (ii) This implication is clear because p 6= 0 and every maximal hyperidealof a hyperring is also a prime hyperideal

(ii) minusrarr (i) Since pR is a prime hyperideal of R ie p is not a unit element of R wehave pR sub R Let I be a hyperideal of R such that pR sube I sub R Since R is a principalhyperideal hyperdomain there exists an element a isin R such that I = aR In addition a cannot be a unit element because I is a proper hyperideal Since R is a unitary hyperring itfollows that p isin I and so p = ab for some b isin R Since pR is a prime hyperideal it follows

91


that p is irreducible and because a is not a unit we have that b is a unit element of R ThuspR = aR = I and therefore pR is a maximal hyperideal

Example 2 ([16]) Consider the set of integers Z and its multiplicative subgroup G = minus1 1Based on Example 1 the Krasnerrsquos construction Z

G is a principal hyperideal hyperdomain Inaddition the prime (also maximal) hyperideals of Z

G have the form lt pG gt where p is a primenumber In Z

G we have lt 0Z gtsublt 2Z gt a chain of prime hyperideals of length 1 Since everynonzero prime hyperideal of Z

G is maximal there does not exist a chain of prime hyperideals of ZG of

length 2 therefore dim ZG = 1

Remark 1 Let R be a non-trivial commutative hyperring By [16] every prime hyperideal of R iscontained in a maximal hyperideal of R (and every maximal hyperideal is prime) Moreover everyprime hyperideal of R contains a minimal prime hyperideal It follows that dimR is equal to thesupremum of lengths of chains P0 sub P1 sub sub Pn of prime hyperideals of R with Pn a maximaland P0 a minimal prime hyperideal Indeed if we have an arbitrary chain of prime hyperideals of Rwith length h like Pprime0 sub Pprime1 sub sub Pprimeh then it is bounded above by the length of a special chain asit follows If Pprime0 is not a minimal prime hyperideal then another prime hyperideal can be insertedbefore it On the other hand if Pprimeh is not a maximal hyperideal of R then another prime hyperidealcan be inserted above it Thus if dimR is finite then

dimR = suphtR M | M is a maximal hyperideal o f R

= suphtRP | P is a prime hyperideal o f R

As a consequence we have the following result

Corollary 1 If R is a local commutative hyperring with the maximal hyperideal M then dimR =htR M

The first property on the height of a hyperideal is highlighted in the next result

Theorem 1 ([16]) Let R be a commutative Noetherian hyperring and let a isin R be a non-unitelement Let P be a minimal prime hyperideal over the principal hyperideal 〈a〉 of R ThenhtRP le 1

Now we can extend Theorem 1 to the case when P is a minimal prime hyperideal overa hyperideal I generated not by one element but by n elements

Theorem 2 ([16]) Let R be a commutative Noetherian hyperring Suppose that I is a properhyperideal of R generated by n elements and P is a minimal prime hyperideal over I ThenhtRP le n

Lemma 2 In a commutative Noetherian hyperring R let I be a hyperideal and P a prime hyperidealof R such that I sube P and htR I = htRP Then P is a minimal prime hyperideal over I

Proof Suppose that P is not a minimal prime hyperideal over I Then by Remark 1 thereexists a prime hyperideal Q of R which is minimal and I sube Q sub P Thus htR I le htRQ lthtRP obtaining a contradiction So P is a minimal prime hyperideal over I

The next theorem states the conditions under which there exists a proper hyperidealof height n and generated by n elements

Theorem 3 ([17]) Let R be a commutative Noetherian hyperring and P a prime hyperideal of Rwith htRP = n Then there exists a proper hyperideal I of R having the following properties

(i) I sube P

92


(ii) I is generated by n elements(iii) htR I = n

We have now all the elements to determine the dimension of a commutative localNoetherian hyperring

Theorem 4 Suppose that R is a commutative local Noetherian hyperring having M as its uniquemaximal hyperideal Then dimR is equal to the smallest number of elements of R that generate anM-primary hyperideal In other words

dimR = mini isin N exista1 a2 ai isin Ri

sumj=1

Raj is an Mminus primary hyperideal

Proof Let

d = mini isin N exist a1 a2 ai isin Ri

sumj=1


By using Corollary 1 and Theorem 2 we have dimR = htR M le d because an M-primary hyperideal must have M as its minimal prime hyperideal On the other hand byusing Lemma 2 and Theorem 3 there exists a hyperideal P of R which has M as a minimalprime hyperideal and which can be generated by dimR = htR M elements In additionevery prime hyperideal of R is contained in M Hence M must be the only one associatedprime hyperideal of P Thus P is an M-primary hyperideal

As a result that there exists an M-primary hyperideal of R which can be generated bydimR elements we have d le dimR Therefore d = dimR and this completes the proof

Definition 16 Let R be a local hyperring of dimension d with M as its unique maximal hyperidealBy regular parameter elements of R we mean a set of d elements of R that generate an M-primaryhyperideal of R

Remark 2 Based on Theorem 4 we conclude that each local hyperring of dimension at least 1possesses a set of regular parameter elements

Example 3 On the set R = 0 1 2 define the hyperaddition + and the multiplication middot by thefollowing tables

+ 0 1 20 0 1 21 1 R 12 2 1 M

middot 0 1 20 0 0 01 0 1 22 0 2 0

Then R is a commutative local Noetherian hypering and M = 0 2 is the only maximal hyperidealof R Thus dimR = htR M and M is an M-primary hyperideal One can check that htR M = 0 sodimR = 0 Therefore this hyperring has no regular parameter elements

Proposition 3 Let R be a local hyperring of dimension d with the unique maximal hyperideal Mand let a1 a2 ad be a set of regular parameter elements for R For any n1 n2 nd isin N wehave that an1

1 an22 and

d form a set of regular parameter elements for R too

Proof The proof is straightforward

93


Proposition 4 Let f R rarr S be a surjective homomorphism of commutative hyperrings Sup-pose that Q1 Q2 Qn P1 P2 Pn are hyperideals of R all of which containing Ker f withr(Qi) = Pi for i = 1 2 n Then

I = Q1 capQ2 cap capQn

is a (minimal) primary decomposition of I if and only if

Ie = Qe1 capQe

2 cap capQen

with r(Qei ) = Pe

i for i = 1 2 n is a (minimal) primary decomposition of Ie


We can conclude that I is a decomposable hyperideal of R (meaning that it has aprimary decomposition) if and only if Ie is a decomposable hyperideal of S Moreover if


where r(Qi) = Pi for i = 1 2 n then we call the set P1 P2 Pn the set of theassociated prime hyperideals related to I and denoted by assR I Using Proposition 4 for thehyperideal Ie of S the related associated prime hyperideals form the set

assS Ie = Pe | P isin assR I = Pe1 Pe

2 Pen

Corollary 2 Let I be a proper hyperideal of the commutative hyperring R Using the canonicalhyperring homomorphism from R to the quotient R

I we conclude that if J is a hyperideal of R suchthat I sube J then J is a decomposable hyperideal of R if and only if J

I is a decomposable hyperideal ofRI and we have

ass RI(

JI) = P

I P isin assR J

Theorem 5 ([17]) Let R be a commutative Noetherian hyperring P a prime hyperideal of R and Ia proper hyperideal of R generated by n elements such that I sube P Then

ht RI

PIle htRP le ht R

I

PI+ n

Proposition 5 Let R be a local commutative hyperring of dimention d with its maximal hyperidealM and let a1 a2 at isin M Then

dimRminus t le dim R(a1 a2 at) le dimR

Moreover dimR(a1 a2 at) = dimRminus t if and only if a1 a2 at are all distinct elementsand form a set of regular parameter elements of R

Proof Using Corollary 1 we have dimR = htR M and therefore

dimR(a1 a2 at) = htR(a1a2at)M(a1 a2 at)

By using Theorem 5 we conclude that


Now let can Rrarr R(a1 a2 at) be the canonical hyperring homomorphism suchthat can(r) = r + (a1 a2 at) Set M = M

(a1a2at)

Suppose that dimR(a1 a2 at) = d minus t Then t le d and by using Theorem 4there exist at+1 at+2 ad isin R(a1 a2 at) where aj = aj + (a1 a2 at) for j =

94


t + 1 d such that (at+1 at+2 ad) is an M-primary hyperideal of R(a1 a2 at)Therefore at+1 at+2 ad isin M Thus

(a1 a2 at at+1 ad)

(a1 a2 at)

is an M-primary hyperideal of R(a1 a2 at) By using Proposition 4 and Corollary 2we conclude that (a1 a2 ad) is an M-primary hyperideal of R It now follows fromTheorem 4 that a1 a2 ad are all distinct and a1 a2 ad is a set of regular parameterelements of R Therefore a1 a2 at are all distinct and form a set of regular parameterelements of R

Now suppose that t le d and there exist at+1 ad isin M such that

a1 a2 at at+1 ad

form a set of regular parameter elements of R Thus (a1 a2 ad) is an M-primaryhyperideal of R Therefore by using Proposition 4 and Corollary 2 (at+1 at+2 ad)is an M-primary hyperideal of R(a1 a2 at) Thus by using Theorem 4 we havedminus t ge dimR But it follows from the first part that dminus t le dimR and so the proof iscomplete

4 Regular Local Hyperrings

The aim of this section is to define the regular local hyperrings For doing this we willfirst prove that in a local Noetherian hyperring R with the unique maximal hyperidealM the R-hypermodule M

M2 is a vectorial hyperspace over the hyperfield RM Then we will

establish a relation between the dimension of the hyperring R and the dimension of thevectorial hyperspace M

M2 Finally we will present a new characterization of the dimensionof the hyperring of fractions

Definition 17 ([18]) Let M be a hypermodule over the commutative hyperring R N be a subhy-permodule of M and I sube M with I 6= 0 We define the hyperideal

(N R I) = r isin R | r middot x isin N f or all x isin I

For any element m isin M we denote (N R m) instead of (N R m) In addition in the specialcase when N = 0 the hyperideal

(0 R I) = r isin R | r middot x = 0 f or all x isin I

is called the annihilator of I and is denoted by AnnR(I) Moreover for any element m isin M wecall the hyperideal (0 R m) the annihilator of the element m

Example 4 Let us continue with Example 3 where R is an R-hypermodule and its hyperideal Mis a subhypermodule We have

AnnR(M) = (0 R M) = r isin R | r middotm = 0 f or all m isin M = M

Moreover for m = 2 we have

(0 R 2) = r isin R | r middot 2 = 0 = 0 2

Proposition 6 Let M be a hypermodule over the commutative hyperring R and I be a hyperidealof R such that I sube AnnR(M) Then M is a hypermodule over R

I

Proof Suppose that r rprime isin R such that r + I = rprime + I We have r minus rprime sube I sube AnnR(M)and so (rminus rprime)m = 0 for any arbitrary element m isin M Thus rm = rprimem Hence we can

95


define a mapping RI timesM minusrarr M such that (r + I m) minusrarr rm It is a routine to check that

M has an RI -hypermodule structure

Proposition 7 Let R be a local hyperring with its maximal hyperideal M and consider thehyperfield F = R

M Let N be a finitely generated R-hypermodule Then the R-hypermodule NMN has

a natural structure as a hypermodule over RM as an F-vectorial hyperspace

Moreover let n1 n2 nt isin N Then the following statements are equivalent

(i) N is generated by n1 n2 nt(ii) The R-hypermodule N

MN is generated by n1 + MN n2 + MN nt + MN(iii) The F-vectorial hyperspace N

MN is generated by n1 + MN n2 + MN nt + MN

Proof Since the R-hypermodule NMN is annihilated by M ie M sube AnnR(

NMN ) by using

Proposition 6 it has a natural structure as hypermodule over RM In addition since F = R

Mis a hyperfield N

MN is also an F-vectorial hyperspaceIt is clear that (i) implies (ii)The R-hypermodule and F-vectorial hyperspace structures of N

MN are related bythe formula

r(n + MN) = (r + M)(n + MN)

for all r isin R and n isin N Thus the equivalence of statements (ii) and (iii) is clear

It remains to prove that (ii) implies (i) Assume that (ii) holds so the R-hypermoduleN

MN is generated by the elements

n1 + MN n2 + MN nt + MN

Let G = Rn1 + Rn2 + + Rnt First we will show that N = G + MN Let n isin N Thenthere exist r1 r2 rt isin R such that

n + MN = r1(n1 + MN) + r2(n2 + MN) + + rt(nt + MN)

Hence nminus sumni=1 rini isin MN It follows that N sube G + MN On the other side it is clear

that G + MN sube N Thus we have N = G + MN Now by using Corollary 29 in [16] weconclude that N = G so N is generated by the elements n1 n2 nt

Theorem 6 The F-vectorial hyperspace NMN in Proposition 7 has finite dimension and the number

of the elements in each minimal generating set for the R-hypermodule N is equal to vdimFN

MN

Proof Since N is a finitely generated R-hypermodule it follows from Proposition 7 thatN

MN is a finitely generated F-vectorial hyperspace Therefore its dimension is finiteLet nprime1 nprime2 nprimep be a minimal generating set for N By Proposition 7 we know that

the setnprime1 + MN nprime2 + MN nprimep + MN

is a generating set for the F-vectorial hyperspace NMN and since nprime1 nprime2 nprimep is a minimal

generating set for N it follows that no proper subset of nprime1 + MN nprime2 + MN nprimep + MNgenerates N

MN Thus the set nprime1 + MN nprime2 + MN nprimep + MN is a basis for the F-vectorial hyperspace N

MN and so vdimFN

MN = p

Note that the R-hypermodule MM2 is annihilated by M If R is a Noetherian hyperring

then by Proposition 6 MM2 has a natural structure as a vectorial hyperspace over the

hyperfield RM In addition using Proposition 7 the dimension of the vectorial hyperspace

MM2 is equal to the number of the elements in an arbitrary minimal generating set for MTherefore we have the following result

96


Theorem 7 Let R be a local Noetherian hyperring with maximal hyperideal M Then

dimR le vdim RM

MM2

Proof The inequality is clear because M is an M-primary hyperideal of the hyperring Rand as mentioned before vdim R

M

MM2 is the number of the elements of an arbitrary minimal

generating set for M Based on Theorem 4 dimR is the smallest number of elements thatgenerate an M-primary hyperideal of R

Definition 18 Let R be a local Noetherian hyperring with maximal hyperideal M Then R iscalled a regular hyperring when dimR = vdim R

M

MM2

Remark 3 For a local Noetherian hyperring R with one maximal hyperideal M and dimR = dwe have the following statements

(i) The dimension of the RM -vectorial hyperspace M

M2 is the number of the elements in each minimalgenerating set for the hyperideal M By using Theorem 4 at least d elements are needed togenerate M and R is a regular hyperring when the hyperideal M can be generated by exactlyd elements

(ii) Suppose that R is a regular hyperring and a1 a2 ad isin M By using Proposition 7 theelements a1 a2 ad generate M if and only if a1 + M2 a2 + M2 ad + M2 in M

M2 forma basis for this R

M -vectorial hyperspace equivalently if and only if a1 + M2 a2 + M2 ad +M2 form a linearly independent set

We conclude this section with a new characterization of the dimension of the hy-perring of fractions RP = Sminus1R where S = R P with P a prime hyperideal of S is amultiplicatively closed subset of R As proved in [1617] RP is a local Noetherian hyperringFirst we recall the main properties of the hyperring of fractions

Proposition 8 ([16]) Let S be a multiplicatively closed subset of a hyperring R

(i) Every hyperideal in Sminus1R is an extended hyperideal(ii) If I is a hyperideal in R then Ie = Sminus1R if and only if I cap S = empty(iii) A hyperideal I is a contracted hyperideal of R if and only if no element of S is a zero divisor in

RI(iv) The prime hyperideals of Sminus1R are in one-to-one correspondence with the prime hyperideals of

R that donrsquot meet S with the correspondence given by Pharr Sminus1P

Theorem 8 Let S be a multiplicatively closed subset of R and P be a prime hyperideal of R suchthat P cap S = empty Then htRP = htSminus1RSminus1P

Proof By Proposition 8 it follows that Sminus1P is a prime hyperideal of Sminus1R Let

P0 sub P1 sub sub Pn = P

be a chain of prime hyperideals of R Again by Proposition 8 it follows that

Pe0 sub Pe

1 sub sub Pen

is a chain of prime hyperideals of Sminus1R with Pen = Pe = Sminus1P and therefore htRP le

htSminus1RSminus1P On the other side if

Q0 sub Q1 sub sub Qn

97


is a chain of prime hyperideals of Sminus1R with Qn = Pe then using Propositions 8 weget that

Qc0 sub Qc

1 sub sub Qcn

is a chain of prime hyperideals of R with Qcn = Pec = P So we have htSminus1RSminus1P le htRP

Therefore it follows that htRP = htSminus1RSminus1P

Combining the previous results we get now the following important consequence

Corollary 3 For a prime hyperideal P of a commutative hyperring R it follows that

htRP = htRP Sminus1P = dim(RP)

In the following we will illustrate the previous results by several examples

Example 5 Let R be a commutative Noetherian hyperring Suppose that there exists a primehyperideal P of R with htP = n and that can be generated by n elements a1 a2 an Considerthe localisation hyperring RP that is a local hyperring According with Theorem 8 and Corollary 3it has dimension n Since the dimension of localization hyperring RP is n the maximal hyperidealof this hyperring ie

PRP = (n

sumi=1

Rai)RP =n

sumi=1

RPai1

can be generated by n elements that are also regular parameter elements it follows that RP is aregular hyperring

Example 6 Let p be a prime number Based on Example 2 we have htZpZ = 1 Since pZ is aprime hyperideal of Z which can be generated by one element it follows from Example 5 that ZpZ isa regular hyperring of dimension 1 and p is a regular parameter element

Example 7 Let R be a principal hyperideal hyperdomain which is not a hyperfield and let M be amaximal hyperideal of R By using Proposition 2 M is a prime hyperideal of R and also principalhyperideal with height 1 Using Example 5 it follows that RM is a regular hyperring of dimension 1

5 Conclusions

In this paper we have to define the regular parameter elements in a commutativelocal Noetherian hyperring R with maximal hyperideal M and present some propertiesrelated to them After investigating some results concerning the quotient hypermodule M

M2

over the hyperfield RM our study has focused on regular local hyperrings We have studied

the relation between the dimension of a commutative local Noetherian hyperring and thedimension of the vectorial hyperspace M

M2 over the hyperfield RM

Our future work will include new results regarding the regular local hyperrings Inparticular we will investigate whether they are hyperdomains or not In addition we willstudy the properties of the hyperideals generated by a subset of the regular parameterelements and the relation between the length of these hyperideals and the dimension ofthe hyperring

Author Contributions Conceptualization HB and IC methodology HB and IC investigationHB and IC writingmdashoriginal draft preparation HB writingmdashreview and editing IC fundingacquisition IC All authors have read and agreed to the published version of the manuscript

Funding The second author acknowledges the financial support from the Slovenian ResearchAgency (research core funding No P1-0285)


98


References1 Kemper G Regular local rings In A Course in Commutative Algebra Graduate Texts in Mathematics Springer Berlin Heildeberg

Germany 2011 Volume 2562 Krasner M Approximation des Corps Values Complets de Caracteristique p p gt 0 Par ceux de Caracteristique Zero Colloque drsquo Algebre

Superieure (Bruxelles Decembre 1956) CBRM Bruxelles Belgium 19573 Krasner M A class of hyperrings and hyperfields Int J Math Math Sci 1983 6 307ndash3124 Rota R Strongly distributive multiplicative hyperrings J Geom 1990 1ndash2 130ndash1385 Ameri R Eyvazi M Hoskova-Mayerova S Superring of Polynomials over a Hyperring Mathematics 2019 7 9026 Mittas JD Sur certaines classes de structures hypercompositionnelles Proc Acad Athens 1973 48 298ndash3187 Connes A Consani C On the notion of geometry over F1 J Algebraic Geom 2011 20 525ndash5578 Jun J Algebraic geometry over hyperrings Adv Math 2018 323 142ndash1929 Viro O Hyperfields for tropical geometry I Hyperfields and dequantization arXiv 2010 arXiv 1006303410 Baker M Bowler N Matroids over partial hyperstructures Adv Math 2019 343 821ndash86311 Jun J Hyperstructures of affine algebraic group schemes J Number Theory 2016 167 336ndash35212 Vahedi V Jafarpour M Aghabozorgi H Cristea I Extension of elliptic curves on Krasner hyperfields Comm Algebra 2019 47

4806ndash482313 Vahedi V Jafarpour M Hoskova-Mayerova S Aghabozorgi H Leoreanu-Fotea V Bekesiene S Derived Hyperstructures

from Hyperconics Mathematics 2020 8 42914 Vahedi V Jafarpour M Cristea I Hyperhomographies on Krasner hyperfields Symmetry 2019 11 144215 Massouros CG Free and cyclic hypermodules Ann Mat Pura Appl 1988 150 153ndash16616 Bordbar H Cristea I Height of prime hyperideals in Krasner hyperrings Filomat 2017 31 6153ndash616317 Bordbar H Cristea I Novak M Height of hyperideals in Noetherian Krasner hyperrings Politehn Univ Bucharest Sci Bull Ser

A Appl Math Phys 2017 79 31ndash4218 Bordbar H Novak M Cristea I A note on the support of a hypermodule J Algebra Appl 2020 19 205001919 Massouros G Massouros C Hypercompositional Algebra Computer Science and Geometry Mathematics 2020 8 133820 Mittas J Espaces vectoriels sur un hypercorps Introduction des hyperspaces affines et Euclidiens Math Balk 1975 5 199ndash21121 Nakassis A Recent results in hyperring and hyperfield theory Int J Math Math Sci 1988 11 209ndash22022 Massouros CG On the theory of hyperrings and hyperfields Algebra Logika 1985 24 728ndash74223 Massouros CG Constructions of hyperfields Math Balk 1991 5 250ndash257

99

mathematics

Article

Fuzzy Multi-Hypergroups

Sarka Hoskova-Mayerova 1 Madeline Al Tahan 2 and Bijan Davvaz 3

1 Department of Mathematics and Physics University of Defence in Brno Kounicova 65 662 10 Brno-stredCzech Republic

2 Department of Mathematics Lebanese International University Beqaa Valley 1803 Lebanonmadelinetahanliuedulb

3 Department of Mathematics Yazd University Yazd 89139 Iran davvazyazdacir Correspondence sarkamayerovaseznamcz


Abstract A fuzzy multiset is a generalization of a fuzzy set This paper aims to combine theinnovative notion of fuzzy multisets and hypergroups In particular we use fuzzy multisetsto introduce the concept of fuzzy multi-hypergroups as a generalization of fuzzy hypergroupsDifferent operations on fuzzy multi-hypergroups are defined and discussed and some results knownfor fuzzy hypergroups are generalized to fuzzy multi-hypergroups

Keywords hypergroup multiset fuzzy multiset fuzzy multi-hypergroup

MSC 20N25 20N20

1 Introduction

In 1934 Frederic Marty [1] defined the concept of a hypergroup as a natural generalization ofa group It is well known that the composition of two elements in a group is an element whereasthe composition of two elements in a hypergroup is a non-empty set The law characterizing sucha structure is called the multi-valued operation or hyperoperation and the theory of the algebraicstructures endowed with at least one multi-valued operation is known as the Hyperstructure TheoryMartyrsquos motivation to introduce hypergroups is that the quotient of a group modulo of any subgroup(not necessarily normal) is a hypergroup Significant progress in the theory of hyperstructureshas been made since the 1970s when its research area was enlarged and different hyperstructureswere introduced (eg hyperrings hypermodules hyperlattices hyperfields etc) Many types ofhyperstructures have been used in different contexts such as automata theory topology cryptographyand code theory geometry graphs and hypergraphs analysis of the convex systems finite groupsrsquocharacter theory theory of fuzzy and rough sets probability theory ethnology and economy [2]An overview of the most important works and results in the field of hyperstructures up to 1993 is givenin the book by Corsini [3] this book was followed in 1994 by the book by Vougiouklis [4] An overviewregarding the applications of hyperstructure theory is given in the book by Corsini and Leoreanu [5]The book by Davvaz and Leoreanu-Fotea [6] deals with the hyperring theory and applications A morerecent book [7] gives an introduction into fuzzy algebraic hyperstructures

A set is a well-defined collection of distinct objects ie every element in a set occurs only onceA generalization of the notion of set was introduced by Yager [8] He introduced the bag (multiset)structure as a set-like object in which repeated elements are significant He discussed operations onmultisets such as intersection and union and he showed the usefulness of the new defined structurein relational databases These new structures have many applications in mathematics and computerscience [9] For example the prime factorization of a positive integer is a multiset in which its elementsare primes (eg 90 has the multiset 2 3 3 5) Moreover the eigenvalues of a matrix (eg the

101


5times 5 lower triangular matrix (aij) with a11 = a22 = minus5 a33 = 1 a44 = a55 = 0 has the multisetminus5minus5 1 0 0) and roots of a polynomial (eg the polynomial (x + 2)2(xminus 1)x3 over the field ofcomplex numbers has the multiset minus2minus2 1 0 0 0) can be considered as multisets

As an extension of the classical notion of sets Zadeh [10] introduced a concept similar to thatof a set but whose elements have degrees of membership and he called it fuzzy set In classical setsthe membership function can take only two values 0 and 1 It takes 0 if the element belongs to theset and 1 if the element does not belong to the set In fuzzy sets there is a gradual assessment ofthe membership of elements in a set which is assigned a number between 0 and 1 (both included)Several applications for fuzzy sets appear in real life We refer to the papers [11ndash13] Yager in [8]generalized the fuzzy set by introducing the concept of fuzzy multiset (fuzzy bag) and he discussed acalculus for them in [14] An element of a fuzzy multiset can occur more than once with possibly thesame or different membership values If every element of a fuzzy multiset can occur at most once wego back to fuzzy sets [15]

In [16] Onasanya and Hoskova-Mayerova defined multi-fuzzy groups and in [1718] the authorsdefined fuzzy multi-polygroups and fuzzy multi-Hv-ideals and studied their properties MoreoverDavvaz in [19] discussed various properties of fuzzy hypergroups Our paper generalizes the workin [161719] to combine hypergroups and fuzzy multisets More precisely it is concerned withfuzzy multi-hypergroups and constructed as follows After the Introduction Section 2 presents somepreliminary definitions and results related to fuzzy multisets and hypergroups that are used throughoutthe paper Section 3 introduces for the first time fuzzy multi-hypergroups as a generalization of fuzzyhypergroups and studies its properties Finally Section 4 defines some operations (eg intersectionselection product etc) on fuzzy multi-hypergroups and discusses them

2 Preliminaries

In this section we present some basic definitions and results related to both fuzzy multisets andhypergroups [2021] that are used throughout the paper

21 Fuzzy Multisets

A multiset is a collection of objects that can be repeated Yager in his paper [8] introduced underthe name bag a structure similar to a set in which repeated elements are allowed He studied someoperations on bags such as intersection union and addition Moreover he introduced the operationof selecting elements from a bag based upon their membership in a set Furthermore he suggested adefinition for fuzzy multisets (fuzzy bags) A multiset (bag) M is characterized by a count functionCM X rarr N where N is the set of natural numbers

In a multiset and unlike a set the multiple occurrences for each of its elements is allowed and itis called multiplicity For example the multiset α β contains two elements α and β each havingmultiplicity 1 whereas the multiset α β β contains two elements α having multiplicity 1 and β

having multiplicity 2 The two multisets α β β and α β are not equal although they are consideredequal as sets

Definition 1 ([1722]) Let X be a set A multiset M drawn from X is represented by a function CM X rarr0 1 2 For each x isin X CM(x) denotes the number of occurrences of x in M

Assume X = a1 ak For a multiset M on X with count function CM the following twoequivalent expressions are used

M = a1n1 aknk

andM = a1 a1︸︷︷︸

n1

ak ak︸︷︷︸nk

102


Here xi has multiplicity ni for all i = 1 k or equivalently CM(xi) = ni for all i = 1 kOne potential useful application for the theory of multisets lies in the field of relational

databases [8]

Example 1 Consider a class of eight students and a relation of students over the scheme (student name grade)given as followsR =(Sam 90) (Nader 85) (Mady 90) (Tala 60) (Joe 70) (Ziad 95) (Lune 85) (Bella 60) Assume we areinterested in finding the grades of students in this class We can take the projection of R on grades and get

Projgrade = 60 70 85 90 95

The set Projgrade does not give the set of grades of all students in the class but it gives the set of different gradesof the students If we need the set of grades of all students in the class then we need to consider the followingmultiset M

M = 60 60 70 85 85 90 90 95 = 602 701 852 902 951

Or equivalently we can say that M is a multiset with count function CM defined as follows CM(60) =

CM(85) = CM(90) = 2 and CM(70) = CM(95) = 1 The notation used here for the multiset M is the same asthat used by Yager in his pioneering paper [8] about multisets

Definition 2 ([23]) Let X be a non-empty set A fuzzy multiset A drawn from X is represented by a functionCMA X rarr Q where Q is the set of all multisets drawn from the unit interval [0 1]

In the above definition the value CMA(x) is a multiset drawn from [0 1] For each x isin X themembership sequence is defined as the decreasingly ordered sequence of elements in CMA(x) and itis denoted by

micro1A(x) micro2

A(x) micropA(x) micro1

A(x) ge micro2A(x) ge ge micro

pA(x)

Fuzzy sets introduced by Zadeh [10] can be considered as a special case of fuzzy multisets by settingp = 1 so that CMA(x) = micro1

A(x)

Example 2 Let X = 0 1 2 3 Then A = (05 05 03 01 01)1 (065 02 02 02)3 is a fuzzymultiset of X with fuzzy count function CMA Or equivalently we can write it as

CMA(0) = CMA(2) = 0 CMA(1) = (05 05 03 01 01) CMA(3) = (065 02 02 02)

Let A B be fuzzy multisets of X with fuzzy count functions CMA CMB respectivelyThen L(x A) = maxj micro

jA(x) 6= 0 and L(x) = L(x A B) = maxL(x A) L(x B) When we

define an operation between two fuzzy multisets the length of their membership sequences should beset as equal In case two membership sequences have different lengths then the shorter sequence isextended with zeros As an illustration we consider the following example

Example 3 Let X = 0 1 2 3 and define the fuzzy multisets A B of X as follows

A = (04 02)0 (05 03 01)1 (065 02 02 02)3

B = (05 03 01 01)1 (05 01)2 (065 02 02)3

In order to define any operations on A B we can rewrite A B as follows

A = (04 02)0 (05 03 01 0)1 (0 0)2 (065 02 02 02)3

B = (0 0)0 (05 03 01 01)1 (05 01)2 (065 02 02 0)3

103


Definition 3 ([24]) Let X be a set and A B be fuzzy multisets of X Then

1 A sube B if and only if CMA(x) le CMB(x) for all x isin X ie microjA(x) le micro

jB(x) j = 1 L(x) for all

x isin X2 A = B if and only if CMA(x) = CMB(x) for all x isin X ie micro

jA(x) = micro


x isin X3 A cap B is defined by micro

jAcapB(x) = micro

jA(x) and micro

jB(x) j = 1 L(x) for all x isin X

4 A cup B is defined by microjAcapB(x) = micro

jA(x) or micro


Example 4 Let X = 0 1 2 3 and define the fuzzy multisets A B C of X as follows

A = (05 03 01 01)1 (065 02 02 02)3

B = (05 03 01 01)1 (05 01)2 (065 02 02 02)3

C = (05 03)0 (05 01)2 (035 03 01)3

Then it is clear that

1 A sube B and A 6= B2 A cap C = (035 02 01)3 and3 A cup C = (05 03)0 (05 03 01 01)1 (05 01)2 (065 03 02 02)3

Definition 4 ([25]) Let X Y be non-empty sets f X rarr Y be a mapping and A B be fuzzy multisets of X Yrespectively Then

1 The image of A under f is denoted by f (A) with fuzzy count function CM f (A) defined as follows For ally isin Y

CM f (A)(y) =

orf (x)=y CMA(x) if fminus1(y) 6= empty

0 otherwise

2 The inverse image of B under f is denoted by fminus1(B) with fuzzy count function CM fminus1(B) defined asCM fminus1(B)(x) = CMB( f (x)) for all x isin X

22 Hypergroups

Let H be a non-empty set and Plowast(H) be the family of all non-empty subsets of H Then amapping Htimes H rarr Plowast(H) is called a binary hyperoperation on H and (H ) is called a hypergroupoid

In the above definition if A and B are two non-empty subsets of H and x isin H then we define

A B =⋃

aisinAbisinB

a b x A = x A and A x = A x

A hypergroupoid (H ) is called a semihypergroup if ldquo is associative ie if x (y z) = (x y) zfor all x y z isin H and is called a quasihypergroup if the reproduction axiom is satisfied ie if x H =

H = H x for all x isin H The couple (H ) is called a hypergroup if it is a semihypergroup and aquasihypergroup A hypergroup (H ) is called commutative if x y = y x for all x y isin H A subsetS of a hypergroup (H ) is called a subhypergroup of H if (S ) is a hypergroup To prove that S is asubhypergroup of H it suffices to show that the reproduction axiom is satisfied for S

Example 5 In the finite hypergroup we can represent the hyperoperation by the Cayley square table in thesame way as for the finite group The only difference is that various places in the table may contain severalelements instead of a single element ie

lowast e a be e a b a ba a e b e bb b e a e a

104


is a hypergroup

Definition 5 ([1726]) Let (H1 1) and (H2 2) be hypergroups and f H1 rarr H2 be a mapping Then f is

1 a homomorphism if f (x 1 y) sube f (x) 2 f (y) for all x y isin H12 a strong homomorphism if f (x 1 y) = f (x) 2 f (y) for all x y isin H13 an isomorphism if f is a bijective strong homomorphism

Example 6 Let H be any non-empty set and define on H as follows

x y = H for all x y isin H

Then (H ) is a hypergroup known as total hypergroup


x y = x y for all x y isin H

Then (H ) is a hypergroup known as biset hypergroup

Example 8 Let Z be the set of integers and define on Z as follows For all x y isin Z

x y =

2Z if x y have same parity

2Z+ 1 otherwise

Then (Z ) is a commutative hypergroup

For more examples and details about hypergroups we refer to the books [3ndash62627] and to thepapers [202128ndash32]

3 Construction of Fuzzy Multi-Hypergroups

Inspired by the definition of the multi-fuzzy group [25] and the fuzzy multi-polygroup [17] weintroduce the concept of the fuzzy multi-hypergroup Further we investigate their properties It iswell known that groups and polygroups [26] are considered as special cases of hypergroups Hencethe results in this section can be considered as more general than that in [1725]

Definition 6 ([7]) Let (H ) be a hypergroup and micro be a fuzzy subset of H Then micro is called a fuzzysubhypergroup of H if for all x y isin H the following conditions hold

1 micro(x) and micro(y) le infzisinxy micro(z)2 for every x a isin H there exists y isin H such that x isin a y and micro(x) and micro(a) le micro(y)3 for every x a isin H there exists z isin H such that x isin z a and micro(x) and micro(a) le micro(z)

Definition 7 Let (H ) be a hypergroup A fuzzy multiset A over H is a fuzzy multi-hypergroup of H if forall x y isin H the following conditions hold

1 CMA(x) and CMA(y) le infzisinxy CMA(z) (or equivalently CMA(x) and CMA(y) le CMA(z) for allz isin x y)

2 for every x a isin H there exists y isin H such that x isin a y and CMA(x) and CMA(a) le CMA(y)3 for every x a isin H there exists z isin H such that x isin z a and CMA(x) and CMA(a) le CMA(z)

Remark 1 It is clear using Definitions 6 and 7 that if (H ) is a hypergroup and micro is a fuzzy subhypergroupof H then micro is a fuzzy multi-hypergroup of H Hence the results of fuzzy subhypergroups are considered aspecial case of our work

105


Remark 2 Let A be a fuzzy multiset over a commutative hypergroup (H ) Then conditions 2 and 3 ofDefinition 7 are equivalent Hence to show that A is a fuzzy multi-hypergroup of H it suffices to show thateither conditions 1 and 2 of Definition 7 are valid or conditions 1 and 3 of Definition 7 are valid

We present different examples on fuzzy multi-hypergroups

Example 9 Let (H ) be the hypergroup defined by the following table

a b

a H a

b a b

It is clear that A = (02 01)a (05 04 04)b is a fuzzy multi-hypergroup of H

Example 10 Let (H1 1) be the hypergroup defined by the following table

1 0 1 2

0 0 1 0 1 H1

1 0 1 0 1 H1

2 H1 H1 2

It is clear that A = (02 01)0 (02 01)1 (05 04 04)2 is a fuzzy multi-hypergroup of H1

Example 11 Let (Z ) be the hypergroup defined in Example 8 It is clear that A with the fuzzy countfunction CMA is a fuzzy multi-hypergroup of Z Where

CMA(x) =

(07 05 05) if x is an even integer

(07 03 02) otherwise

Proposition 1 Let (H ) be a hypergroup and A be a fuzzy multi-hypergroup of H Then the followingassertions are true

1 CMA(z) ge CMA(x1) and and CMA(xn) for all z isin x1 xn and n ge 22 CMA(z) ge CMA(x) for all z isin xn

Proof

bull Proof of 1 By mathematical induction on the value of n CMA(z) ge CMA(x1) and and CMA(xn)

for all z isin x1 xn is true for n = 2 Assume that CMA(z) ge CMA(x1) and and CMA(xn) forall z isin x1 xn and let zprime isin x1 xn xn+1 Then there exists x isin x1 xn such that zprime isinx xn+1 Having A a fuzzy multi-hypergroup implies that CMA(zprime) ge CMA(x) and CMA(xn+1)And using our assumption that CMA(x) ge CMA(x1)and andCMA(xn) implies that our statementis true for n + 1

bull Proof of 2 The proof follows from 1 by setting xi = x for all i = 1 n

Example 12 Let (H ) be any hypergroup and a isin H be a fixed element We define a fuzzy multiset A of Hwith fuzzy count function CMA as CMA(x) = CMA(a) for all x isin H Then A is a fuzzy multi-hypergroup ofH (the constant fuzzy multi-hypergroup)

Remark 3 Let (H ) be a hypergroup Then we can define at least one fuzzy multi-hypergroup of H which ismainly the one that is described in Example 12

106


Proposition 2 Let (H ) be the biset hypergroup and A be a fuzzy multiset of H Then A is a fuzzymulti-hypergroup of H

Proof Let A be a fuzzy multiset of H Since (H ) is a commutative hypergroup it suffices to showthat conditions 1 and 2 of Definition 7 are satisfied (See Remark 2) For condition 1 let x y isin H andz isin x y = x y It is clear that CMA(z) ge CMA(x) and CMA(y) For condition 2 let a x isin H Thenthere exists y = x isin H such that x isin a y and CMA(y) = CMA(x) ge CMA(a) and CMA(x) ThereforeA is a fuzzy multi-hypergroup of H

Proposition 3 Let (H ) be the total hypergroup and A be a fuzzy multiset of H Then A is a fuzzymulti-hypergroup of H if and only if A is the fuzzy multiset described in Example 12

Proof If A is the fuzzy multiset described in Example 12 then it is clear that A is a fuzzymulti-hypergroup of H Let A be a fuzzy multi-hypergroup of H and a isin H For all x isin H wehave x isin a a = H and a isin x x The latter and having A a fuzzy multi-hypergroup of H implies thatCMA(x) ge CMA(a) and CMA(a) ge CMA(x) Thus CMA(x) = CMA(a) for all x isin H

Notation 1 Let (H ) be a hypergroup A be a fuzzy multiset of H and CMA(x) =

(micro1A(x) micro2

A(x) micropA(x)) We say that CMA(x) gt 0 if micro1

A(x) gt 0

Definition 8 Let (H ) be a hypergroup and A be a fuzzy multiset of H Then A = x isin X CMA(x) gt 0

Proposition 4 Let (H ) be a hypergroup and A be a fuzzy multi-hypergroup of H Then A is either theempty set or a subhypergroup of H

Proof Let a isin A 6= empty We need to show that the reproduction axiom is satisfied for A Weshow that a A = A and A a = A is done similarly For all x isin A and z isin a x we haveCMA(z) ge CMA(a) and CMA(x) gt 0 The latter implies that z isin A and hence A a sube A Moreoverfor all x isin A Condition 2 of Definition 7 implies that there exist y isin H such that x isin a yand CMA(y) ge CMA(x) and CMA(a) gt 0 The latter implies that y isin A and x isin a A ThusA sube a A

Definition 9 Let (H ) be a hypergroup and A B be fuzzy multisets of H Then A B is defined by thefollowing fuzzy count function

CMAB(x) = orCMA(y) and CMB(z) x isin y z

Theorem 1 Let (H ) be a hypergroup and A be a fuzzy multiset of H If A is a fuzzy multi-hypergroup of Hthen A A = A

Proof Let z isin H Then CMA(z) ge CMA(x) and CMA(y) for all z isin x y The latter implies thatCMA(z) ge orCMA(x) and CMB(y) z isin x y ge CMAA(z) Thus A A sube A Having (H ) ahypergroup and A a fuzzy multi-hypergroup of H implies that for every x isin H there exist y isin H suchthat x isin x y and CMA(y) ge CMA(x) Moreover we have CMAA(x) = orCMA(y) and CMB(z) x isiny z ge CMA(x) and CMA(y) = CMA(x) Thus A sube A A


(micro1A(x) micro2

A(x) micropA(x)) We say that CMA(x) ge (t1 tk) if p ge k and microi

A(x) ge ti for all i = 1 kIf CMA(x) (t1 tk) and (t1 tk) CMA(x) then we say that CMA(x) and (t1 tk) arenot comparable

107


In [19] Davvaz studied fuzzy hypergroups and proved some important results about them relatedto level subhypergroups of fuzzy hypergroups In what follows we do suitable changes to extend theresults of [19] to fuzzy multi-hypergroups

Theorem 2 Let (H ) be a hypergroup A a fuzzy multiset of H with fuzzy count function CM and t =

(t1 tk) where ti isin [0 1] for i = 1 k and t1 ge t2 ge ge tk Then A is a fuzzy multi-hypergroup of Hif and only if CMt is either the empty set or a subhypergroup of H

Proof Let CMt be a subhypergroup of H and x y isin H By setting t0 = CM(x) and CM(y) we getthat x y isin CMt0 Having CMt0 a subhypergroup of H implies that for all z isin x y CM(z) get0 = CM(x) and CM(y) We prove condition 2 of Definition 7 and condition 3 is done similarlyLet a x isin H and t0 = CM(x) and CM(a) Then a x isin CMt0 Having CMt0 a subhypergroup of Himplies that a CMt0 = CMt0 The latter implies that there exist y isin CMt0 such that x isin a y ThusCM(y) ge t0 = CM(x) and CM(y)

Conversely let A be a fuzzy multi-hypergroup of H and CMt 6= empty We need to show thatCMt = a CMt = CMt a for all a isin CMt We prove that CMt = a CMt and CMt = CMt a is donesimilarly Let x isin CMt Then CM(z) ge CM(x) and CM(a) ge t for all z isin a x The latter implies thatz isin CMt Thus a CMt sube CMt Let x isin CMt Since A is a fuzzy multi-hypergroup of H it followsthat there exist y isin H such that x isin a y and CM(y) ge CM(x) and CM(a) ge t The latter implies thaty isin CMt and hence CMt sube a CMt

Proposition 5 Let (H ) be a hypergroup A a fuzzy multiset of H with fuzzy count function CM andt = (t1 tk) s = (s1 sn) where ti si isin [0 1] for i = 1 max(k n) and t1 ge t2 ge ge tks1 ge s2 ge ge sn If t lt s and CMt = CMs then there exist no x isin H such that t le CM(x) lt s

Proof Let CMt = CMs and suppose that there exist x isin H such that t le CM(x) lt s Then x isin CMt

and x isin CMs which contradicts the given

Proposition 6 Let (H ) be a hypergroup and S be a subhypergroup of H Then S = CMt for somet = (t1 tk) where ti isin [0 1] for i = 1 k t1 6= 0 and t1 ge t2 ge ge tk

Proof Let t = (t1 tk) where ti isin [0 1] for i = 1 k and define the fuzzy multiset A of Has follows

CM(x) =

t if x isin S

0 otherwise

It is clear that S = CMt We still need to prove that CM is a fuzzy multi-hypergroup of H UsingTheorem 2 it suffices to show that CMα 6= empty is a subhypergroup of H for all α = (a1 as) withai isin [0 1] for i = 1 s One can easily see that

CMα =

H if α = 0

S if 0 lt α le t

empty if (α gt t) or (α and t are not comparable)

Thus CMα is either the empty set or a subhypergroup of H

4 Operations on Fuzzy Multi-Hypergroups

In this section we define some operations on fuzzy multi-hypergroups study them and presentsome examples

108


Proposition 7 Let (H ) be a hypergroup and A B be fuzzy multisets of H If A and B are fuzzymulti-hypergroups of H and one of them is the constant fuzzy multi-hypergroup then A cap B is a fuzzymulti-hypergroup of H

Proof We prove conditions of Definition 7 are satisfied for A cap B (1) Let x y isin H and z isinx y Then CMAcapB(z) = CMA(z) and CMB(z) Having A B fuzzy multi-hypergroups of H impliesthat CMA(z) ge CMA(x) and CMA(y) and CMB(z) ge CMB(x) and CMB(y) The latter implies thatCMAcapB(z) ge CMA(x) and CMA(y) and CMB(x) and CMB(y) = CMAcapB(x) and CMAcapB(y) (2) Withoutloss of generality let B be the constant fuzzy multiset of H with CMB(x) = α for all x isin H Leta x isin H Then there exist y isin H such that x isin a y and CMA(y) ge CMA(x) and CMA(a) The latterimplies that CMAcapB(y) = CMA(y)and α ge CMA(x)andCMA(a)and α = (CMA(x)and α)and (CMA(a)and α) =

CMAcapB(x) and CMAcapB(a) (3) is done in a similar way to (2)

Example 13 Let (H ) be the hypergroup defined in Example 9 and A B be fuzzy multisets of H defined as

A = (02 01)a (05 04 04)b B = (07 005 005)a (07 005 005)b

Since A is a fuzzy multi-hypergroup of H and B is a constant fuzzy multi-hypergroup of H it follows thatA cap B = (02 005)a (05 005 005)b is a fuzzy multi-hypergroup of H

Definition 10 Let H be any set and A B be fuzzy multisets of H with fuzzy count functions CMA CMBrespectively Then the fuzzy multiset A

⊎B is given as follows For all x isin H

CM(x) =CMA(x) + CMB(x)

2

Example 14 Let H = a b A B be fuzzy multisets of H given as

A = (08 04 04)a (01 01)b B = (06 04 04 04)a (03 01 01)b

Then A⊎

B = (07 04 04 02)a (02 01 005)b

Proposition 8 Let (H ) be a hypergroup and A B be fuzzy multisets of H If A and B are fuzzymulti-hypergroups of H and A or B is constant then A

⊎B is a fuzzy multi-hypergroup of H

Proof Without loss of generality let B be the constant fuzzy multiset of H with CMB(x) = α forall x isin H We prove that the conditions of Definition 7 are satisfied for A

⊎B (1) Let x y isin H and

z isin x y Having CMA(z) ge CMA(x) and CMA(y) implies that CMA(z)+α2 ge (CMA(x)andCMA(y))+α

2 We

get that CM(z) = CMA(x)+α2 ge CMA(x)+α

2 and CMA(y))+α2 = CM(x) and CM(y) (2) Let a x isin H Then

there exist y isin H such that x isin a y and CMA(y) ge CMA(x) and CMA(a) The latter implies thatCM(y) = CMA(y)+α

2 ge (CMA(x)andCMA(a))+α2 = CMA(x)+α

2 and CMA(a))+α2 = CM(x) and CM(a) (3) is done in

a similar way to (2)

Example 15 In Example 13 A⊎

B = (045 0075 0025)a (06 0225 0225)b is a fuzzymulti-hypergroup of H

Definition 11 Let H be a non-empty set and A be a fuzzy multiset of H We define Aprime the complement of Ato be the fuzzy multiset defined as For all x isin H

CMAprime(x) = (1minus micropA(x) 1minus micro1

A(x)) when CMA(x) = (micro1A(x) micro2

A(x) micropA(x))

109


Remark 4 Let (H ) be a hypergroup and A be the constant fuzzy multi-hypergroup of H defiend inExample 12 Then Aprime is also a fuzzy multi-hypergroup of H

We present an example of a (non-constant) fuzzy multiset A of H where A and Aprime are both fuzzymulti-hypergroups of H

Example 16 Let H = a b (H ) be the biset hypergroup on H and A be a fuzzy multiset of H defined asA = (04 03 03)a (02 01)b Proposition 2 asserts that A and Aprime = (07 07 06)a (09 08)bare fuzzy multi-hypergroups of H

Remark 5 Let (H ) be a hypergroup and A be a fuzzy multi-hypergroup of H Then Aprime is not necessary afuzzy multi-hypergroup of H

The following example is an illustration for Remark 5

Example 17 Let (H ) be the hypergroup defined in Example 9 and A be the fuzzy multi-hypergroup of Hdefined as A = (02 01)a (05 04 04)b Then Aprime = (09 08)a (06 06 05)b is not a fuzzymulti-hypergroup of H as b isin a a and CMAprime(b) CMAprime(a)

Definition 12 Let X be any set S sube X and A a fuzzy multiset of X with fuzzy count function CMA Thenthe selection operation otimes is defined by the fuzzy multiset Aotimes S with the fuzzy count function CM as follows

CM(x) =

CMA(x) if x isin S

0 otherwise

Proposition 9 Let (H ) be a hypergroup and S be a subhypergroup of H If A is a fuzzy multi-hypergroupof S then Aotimes S is a fuzzy multi-hypergroup of H

Proof Let z isin x y If x isin S or y isin S then CM(z) ge 0 = CM(x) and CM(y) If x y isin S then z isin x yHaving A a fuzzy multi-hypergroup of S implies that CMA(z) ge CMA(x) and CMA(y) The latterimplies that CM(z) = CMA(z) ge CMA(x) and CMA(y) = CM(x) and CM(y) We prove condition 2 ofDefinition 7 and condition 3 is done similarly Let a x isin H If a isin S or x isin S then for all y isin H withx isin a y CM(y) ge 0 = CM(x) and CM(a) If a x isin S then there exist y isin S such that x isin a y andCMA(y) ge CMA(x) and CMA(a) The latter implies that CM(y) ge CM(x) and CM(a)

Definition 13 Let X be any set S sube X and A a fuzzy multiset of X with fuzzy count function CMA Thenthe selection operation is defined by the fuzzy multiset Aotimes S with the fuzzy count function CM

CM(x) =

CMA(x) if x isin S

0 otherwise

Proposition 10 Let (H ) be a hypergroup and S sub H with H minus S a subhypergroup of H If A is a fuzzymulti-hypergroup of H minus S then A S is a fuzzy multi-hypergroup of H

Proof A S is given by the fuzzy count function CM where

CM(x) =

0 if x isin S

CMA(x) otherwise=

0 if x isin H minus S

CMA(x) otherwise

One can easily see that A S = Aotimes (H minus S) Proposition 9 completes the proof

110


Proposition 11 (H1 1) (H2 2) be hypergroups with fuzzy multisets A B respectively If A and B arefuzzy multi-hypergroups of H1 and H2 respectively then Atimes B is a fuzzy multi-hypergroup of the productionalhypergrpup (H1 times H2 ) where for all (x y) isin H1 times H2 CMAtimesB(x y) = CMA(x) and CMB(y)

Proof Let (x3 y3) isin (x1 y1) (x2 y2) Then x3 isin x1 x2 and y3 isin y1 y2 Having A Bfuzzy multi-hypergroups of H1 H2 respectively implies that CMA(x3) ge CMA(x1) and CMA(x2) andCMB(y3) ge CMB(y1) and CMB(y2) We get now

CMAtimesB(x3 y3) = CMA(x3) and CMB(y3) ge CMA(x1) and CMA(x2) and CMB(y1) and CMB(y2)

The latter implies that CMAtimesB(x3 y3) ge CMAtimesB(x1 y1) and CMAtimesB(x2 y2) We now prove condition2 of Definition 7 and condition 3 is done similarly Let (x y) (a b) isin H1 times H2 Havingx a isin H1 y b isin H2 implies that there exist z isin H1 w isin H2 such that x isin a z y isin b wand CMA(z) ge CMA(x) and CMA(a) CMB(w) ge CMB(y) and CMB(b) We get that CMAtimesB(z w) =

CMA(z) and CMB(w) ge CMA(x) and CMA(a) and CMB(w) ge CMB(y) and CMB(b) = CMAtimesB(x y) andCMAtimesB(a b) The latter implies that there exist (z w) isin H1 H2 such that (x y) isin (a b) (z w)

and CMAtimesB(z w) ge CMAtimesB(x y) and CMAtimesB(a b)

Corollary 1 (Hi i) be a hypergroup with fuzzy multiset Ai for i = 1 n If Ai is a fuzzy multi-hypergroupof Hi then A1 times times An is a fuzzy multi-hypergroup of the productional hypergroup (H1 times times Hn )where for all (x1 xn) isin H1 times times Hn CMA1timestimesAn(x1 xn) = CMA1(x1) and and CMAn(xn)

Proof The proof follows by using mathematical induction and Proposition 11

Example 18 Let (H ) be the hypergroup defined in Example 9 and (J ) be the biset hypergroup on the setc d Then the productional hypergroup (H times J bull) is given by the following table

bull (a c) (a d) (b c) (b d)

(a c) (a c) (a c) (a d) (a c) (b c) H times J

(a d) (a c) (a d) (a d) H times J (a d) (b d)(b c) (a c) (b c) H times J (b c) (b c) (b d)(b d) H times J (a d) (b d) (b c) (b d) (b d)

Since A = (02 01)a (05 04 04)b is a fuzzy multi-hypergroup of H and B =

(03 005)c (02 02 01)d is a fuzzy multi-hypergroup of J Then

Atimes B = (02 005)(a c) (02 01)(a d) (03 005)(b c) (02 02 01)(b d)

is a fuzzy multi-hypergroup of H times J

The following propositions (Propositions 12 and 13) deal with the strong homomorphic imageand pre-image of a fuzzy multi-hypergroup

Proposition 12 Let (H1 1) (H2 2) be hypergroups A B be fuzzy multisets of H1 H2 respectively andf H1 rarr H2 be a strong homomorphism If A is a fuzzy multi-hypergroup of H1 then f (A) is a fuzzymulti-hypergroup of H2

Proof Let y1 y2 isin H2 and y3 isin y1 2 y2 If fminus1(y1) = empty or fminus1(y2) = empty then CM f (A)(y1) = 0 orCM f (A)(y2) = 0 We get that CM f (A)(y3) ge 0 = CM f (A)(y1) and CM f (A)(y2) If fminus1(y1) 6= empty andfminus1(y2) 6= empty then there exist x1 x2 isin H1 such that CMA(x1) =

orf (x)=y1

CMA(x) and CMA(x2) =

111


orf (x)=y2

CMA(x) Having f a homomorphism implies that y3 isin f (x1) 2 f (x2) = f (x1 1 x2) Thelatter implies that there exists x3 isin x1 x2 such that y3 = f (x3) Since A is a fuzzy multi-hypergroup ofH1 it follows that CM f (A)(y3) ge CMA(x3) ge CMA(x1)andCMA(x2) = CM f (A)(y1)andCM f (A)(y2) Weprove now condition 2 of Definition 7 and condition 3 is done similarly Let y b isin H2 If fminus1(y) = emptyor fminus1(b) = empty then for all z isin H2 such that y isin b 2 z CM f (A)(z) ge 0 = CM f (A)(y) and CM f (A)(b)If fminus1(y) 6= empty and fminus1(b) 6= empty then there exist x1 a isin H1 such that CMA(x1) =

orf (x)=y CMA(x)

and CMA(a) =or

f (x)=b CMA(x) Having A a fuzzy multi-hypergroup of H1 implies that there existx2 isin H with x1 isin x2 1 a and CMA(x2) ge CMA(x1) and CMA(a) Since f is a strong homomorphismit follows that y = f (x1) isin f (x2) 2 b and CM f (A)( f (x2)) ge CMA(x2) ge CMA(x1) and CMA(a) =

CM f (A)(y) and CM f (A)(b)

We can use Proposition 12 to prove Proposition 9

Corollary 2 Let (H ) be a hypergroup and S be a subhypergroup of H If A is a fuzzy multi-hypergroup of Sthen Aotimes S is a fuzzy multi-hypergroup of H

Proof Let f Srarr H be the inclusion map defined by f (x) = x for all x isin S One can easily see thatCM f (A) is the fuzzy count function of Aotimes S

Proposition 13 Let (H1 1) (H2 2) be hypergroups A B be fuzzy multisets of H1 H2 respectively andf H1 rarr H2 be a surjective strong homomorphism If B is a fuzzy multi-hypergroup of H2 then fminus1(B) is afuzzy multi-hypergroup of H1

Proof Let x1 x2 isin H1 and x3 isin x1 1 x2 Then CM fminus1(B)(x3) = CMB( f (x3)) Having f (x3) isin f (x1 1

x2) = f (x1) f (x2) implies that CM fminus1(B)(x3) = CMB( f (x3)) ge CMB( f (x1)) and CMB( f (x2)) =

CM fminus1(B)(x1) and CM fminus1(B)(x2) We prove now condition 2 of Definition 7 and condition 3 is donesimilarly Let x a isin H1 Having y = f (x) b = f (a) isin H2 and B a fuzzy multi-hypergroup of H2

implies that there exist z isin H2 such that y isin b 2 z and CMB(z) ge CMB(y) and CMB(b) Since f is asurjective strong homomorphism it follows that there exist w isin H1 such that f (w) = z and x isin z 1 wWe get now that CM fminus1(B)(z) = CMB(z) ge CMB(y) and CMB(b) = CM fminus1(B)(x) and CM fminus1(B)(w)

5 Conclusions

In this paper a new link between algebraic hyperstructures and fuzzy multisets was initiated andas a result fuzzy multi-hypergroups were defined and studied In particular different operations onfuzzy multi-hypergroups were defined and studied and several results and examples were obtainedThe foundations that we made through this paper can be used to get an insight into other types ofhyperstructures As a result different real life problems involving the concept of the fuzzy multisetcan be dealt with from a different perspective

Author Contributions Conceptualization SH-M MAT and BD methodology SH-M MAT and BDinvestigation SH-M MAT and BD resources SH-M MAT and BD writingmdashoriginal draft SH-MMAT and BD writingmdashreview and editing SH-M MAT and BD project administration SH-M fundingacquisition SH-M All authors have read and agreed to the published version of the manuscript

Funding The APC was funded by by the Ministry of Defence in the Czech Republic

Acknowledgments The work presented in this paper was supported within the project for development ofbasic and applied research developed in the long term by the departments of theoretical and applied bases FMTsupported by the Ministry of Defence in the Czech Republic


112


References

1 Marty F Sur une generalization de la notion de group In Proceedings of the 8th Congress on MathmaticsScandenaves Stockholm Sweden 14ndash18 August 1934 pp 45ndash49

2 Antampoufis N Hoskova-Mayerova S A Brief Survey on the two Different Approaches of FundamentalEquivalence Relations on Hyperstructures Ratio Mathematica 2017 33 47ndash60 [CrossRef]

3 Corsini P Prolegomena of Hypergroup Theory 2nd ed Udine Tricesimo Italy 19934 Vougiouklis T Hyperstructures and Their Representations Hadronic Press Monographs Palm Harbor FL

USA 19945 Corsini P Leoreanu V Applications of Hyperstructures Theory Advances in Mathematics Kluwer Academic

Publisher Dordrecht The Netherlands 20036 Davvaz B Leoreanu-Fotea V Hyperring Theory and Applications International Academic Press Cambridge

MA USA 20077 Davvaz B Cristea I Fuzzy Algebraic Hyperstructures Studies in Fuzziness and Soft Computing 321 Springer

International Publishing Cham Switzerland 2015 [CrossRef]8 Yager R On the theory of bags Int J Gen Syst 1987 13 23ndash37 [CrossRef]9 Onasanya BO Hoskova-Mayerova S Results on Functions on Dedekind Multisets Symmetry 2019 11

1125 [CrossRef]10 Zadeh LA Fuzzy sets Inform Control 1965 8 338ndash353 [CrossRef]11 Mei Y Peng J Yang J Convex aggregation operators and their applications to multi-hesitant fuzzy

multi-criteria decision-making Information 2018 9 207 [CrossRef]12 Joshi DK Beg I Kumar S Hesitant probabilistic fuzzy linguistic sets with applications in multi-criteria

group decision making problems Mathematics 2018 6 47 [CrossRef]13 Yaqoob N Gulistan M Kadry S Wahab HA Complex intuitionistic fuzzy graphs with application in

cellular network provider companies Mathematics 2019 7 35 [CrossRef]14 Miyamoto S Fuzzy Multisets and Their Generalizations Multiset Processing Springer Berlin Germany 2001

pp 225ndash23515 Onasanya BO Hoskova-Mayerova S Some Topological and Algebraic Properties of alpha-level Subsetsrsquo

Topology of a Fuzzy Subset Analele St Univ Ovidius Constanta 2018 26 213ndash227 [CrossRef]16 Onasanya BO Hoskova-Mayerova S Multi-fuzzy group induced by multisets Ital J Pure Appl Math

2019 41 597ndash60417 Al Tahan M Hoskova-Mayerova S Davvaz B Fuzzy multi-polygroups J Intell Fuzzy Syst 2019

[CrossRef]18 Al Tahan M Hoskova-Mayerova S Davvaz B Some results on (generalized) fuzzy multi-Hv-ideals of

Hv-rings Symmetry 2019 11 1376 [CrossRef]19 Davvaz B Fuzzy Hv-groups Fuzzy Sets Syst 1999 101 191ndash195 [CrossRef]20 Cristea I Several aspects on the hypergroups associated with n-ary relations Analele Stiint Univ Ovidius

Constanta Ser Mat 2009 17 99ndash11021 Yaqoob N Cristea I Gulistan M Left almost polygroups Ital J Pure Appl Math 2018 39 465ndash47422 Jena SP Ghosh SK Tripathi BK On theory of bags and lists Inform Sci 2011 132 241ndash254 [CrossRef]23 Shinoj TK John SJ Intutionistic fuzzy multisets Int J Eng Sci Innov Technol (IJESIT) 2013 2 1ndash2424 Syropoulos A Mathematics of Multisets Multiset Processing Springer Berlin Germany 2001 pp 347ndash35825 Shinoj TK Baby A John SJ On some algebraic structures of fuzzy multisets Ann Fuzzy Math Inform

2015 9 77ndash9026 Davvaz B Polygroup Theory and Related Systems World Scientific Publishing Co Pte Ltd Hackensack NJ

USA 201327 Davvaz B Semihypergroup Theory ElsevierAcademic Press London UK 201628 Leoreanu-Fotea V Rosenberg I Davvaz B Vougiouklis T A new class of n-ary hyperoperations Eur J

Comb 2015 44 265ndash273 [CrossRef]29 Cristea I Fuzzy Subhypergroups Degree J Mult Valued Log Soft Comput 2016 27 75ndash8830 Cristea I Hoskova S Fuzzy pseudotopological hypergroupoids Iran J Fuzzy Syst 2009 6 11ndash19

113


31 Vougiouklis T On the hyperstructure theory Southeast Asian Bull Math 2016 40 603ndash62032 De Salvo M Fasino D Freni D Faro GL On hypergroups with a β-class of finite height Symmetry 2020

12 168 [CrossRef]


114

mathematics

Article

Sequences of Groups Hypergroups and Automata of LinearOrdinary Differential Operators

Jan Chvalina 1 Michal Novaacutek 1 Bedrich Smetana 2 and David Stanek 1

Citation Chvalina J Novaacutek M

Smetana B Stanek D Sequences of

Groups Hypergroups and Automata

of Linear Ordinary Differential

Operators Mathematics 2021 9 319

httpsdoiorg103390math9040319

Academic Editor Christos G

Massouros


Accepted 2 February 2021

Published 5 February 2021




nal affiliations








40)

1 Department of Mathematics Faculty of Electrical Engineeering and Communication Brno University ofTechnology Technickaacute 8 616 00 Brno Czech Republic chvalinafeecvutbrcz (JC)xstanek41studfeecvutbrcz (DS)

2 Department of Quantitative Methods University of Defence in Brno Kounicova 65662 10 Brno Czech Republic bedrichsmetanaunobcz

Correspondence novakmfeecvutbrcz Tel +420-541146077

Abstract The main objective of our paper is to focus on the study of sequences (finite or countable)of groups and hypergroups of linear differential operators of decreasing orders By using a suitableordering or preordering of groups linear differential operators we construct hypercompositionalstructures of linear differential operators Moreover we construct actions of groups of differentialoperators on rings of polynomials of one real variable including diagrams of actionsndashconsidered asspecial automata Finally we obtain sequences of hypergroups and automata The examples wechoose to explain our theoretical results with fall within the theory of artificial neurons and infinitecyclic groups

Keywords hyperstructure theory linear differential operators ODE automata theory

1 Introduction

This paper discusses sequences of groups hypergroups and automata of linear dif-ferential operators It is based on the algebraic approach to the study of linear ordinarydifferential equations Its roots lie in the work of Otakar Boruvka a Czech mathemati-cian who tied the algebraic geometrical and topological approaches and his successorFrantišek Neuman who advocated the algebraic approach in his book [1] Both of them(and their students) used the classical group theory in their considerations In severalpapers published mainly as conference proceedings such as [2ndash4] the existing theorywas extended by the use of hypercompositional structures in place of the usual algebraicstructures The use of hypercompositional generalizations has been tested in the automatatheory where it has brought several interesting results see eg [5ndash8] Naturally this ap-proach is not the only possible one For another possible approach investigations ofdifferential operators by means of orthognal polynomials see eg [910]

Therefore in this present paper we continue in the direction of [24] presenting resultsparallel to [11] Our constructions no matter how theoretical they may seem are motivatedby various practical issues of signal processing [12ndash16] We construct sequences of groupsand hypergroups of linear differential operators This is because in signal processing (butalso in other real-life contexts) two or more connecting systems create a standing highersystem characteristics of which can be determined using characteristics of the originalsystems Cascade (serial) and parallel connecting of systems of signal transfers are used inthis Moreover series of groups motivated by the Galois theory of solvability of algebraicequations and the modern theory of extensions of fields are often discussed in literatureNotice also paper [11] where the theory of artificial neurons used further on in someexamples has been studied

Another motivation for the study of sequences of hypergroups and their homomor-phisms can be traced to ideas of classical homological algebra which comes from the

115


algebraic description of topological spaces A homological algebra assigns to any topo-logical space a family of abelian groups and to any continuous mapping of topologicalspaces a family of group homomorphisms This allows us to express properties of spacesand their mappings (morphisms) by means of properties of the groups or modules or theirhomomorphisms Notice that a substantial part of homology theory is devoted to the studyof exact short and long sequences of the above mentiones structures

2 Sequences of Groups and Hypergroups Definitions and Theorems21 Notation and Preliminaries

It is crucial that one understands the notation used in this paper Recall that westudy by means of algebra linear ordinary differential equations Therefore our notationwhich follows the original model of Boruvka and Neuman [1] uses a mix of algebraic andfunctional notation

First we denote intervals by J and regard open intervals (bounded or unbounded)Systems of functions with continuous derivatives of order k on J are denoted by Ck(J) fork = 0 we write C(J) instead of C0(J) We treat Ck(J) as a ring with respect to the usualaddition and multiplication of functions We denote by δij the Kronecker delta i j isin N ieδii = δjj = 1 and δij = 0 whenever i 6= j by δij we mean 1minus δij Since we will be usingsome notions from the theory of hypercompositional structures recall that by P(X) onemeans the power set of X while (P)lowast(X) means P(X) empty

We regard linear homogeneous differential equations of order n ge 2 with coefficientswhich are real and continuous on J andndashfor convenience reasonsndashsuch that p0(x) gt 0 forall x isin J ie equations

y(n)(x) + pnminus1(x)y(nminus1)(x) + middot middot middot+ p0(x)y(x) = 0 (1)

By An we adopting the notation of Neuman [1] mean the set of all such equations

Example 1 The above notation can be explained on an example taken from [17] in which Neumanconsiders the third-order linear homogeneous differential equation

yprimeprimeprime(x)minus qprime1(x)q1(minusx)

yprimeprime(x) + (q1(x)minus 1)2yprime(x)minus qprime1(x)q1(x)

y(x) = 0

on the open interval J isin R One obtains this equation from the system

yprime1 = y2

yprime2 = minusy1 + q1(x)y3

yprime3 = minusq1(x)y2

Here q1 isin C+(J) satisfies the condition q1(x) 6= 0 on J In the above differential equation we

have n = 3 p0(x) = minus qprime1(x)q1(x) p1(x) = (q1(x)minus 1)2 and p2(x) = minus qprime1(x)

q1(minusx) It is to be noted thatthe above three equations form what is known as set of global canonical forms for the third-orderequation on the interval J

Denote Ln(pnminus1 p0) Cn(J) rarr Cn(J) the above linear differential operator de-fined by

Ln(pnminus1 p0)y(x) = y(n)(x) +nminus1

sumk=0

pk(x)y(k)(x) (2)

where y(x) isin Cn(J) and p0(x) gt 0 for all x isin J Further denote by LAn(J) the set of allsuch operators ie

LAn(J) = L(pnminus1 p0) | pk(x) isin C(J) p0(x) gt 0 (3)

116


By LAn(J)m we mean subsets of LAn(J) such that pm isin C+(J) ie there is pm(x) gt 0for all x isin J If we want to explicitly emphasize the variable we write y(x) pk(x) etcHowever if there is no specific need to do this we write y pk etc Using vector notation~p(x) = (pnminus1(x) p0(x)) we can write

Ln(~p)y = y(n) +nminus1

sumk=0

pky(k) (4)

Writing L(~p) isin LAn(J) (or L(~p) isin LAn(J)m) is a shortcut for writing Ln(~p)y isin LAn(J)(or Ln(~p)y isin LAn(J)m)

On the sets of linear differential operators ie on sets LAn(J) or their subsetsLAn(J)m we define some binary operations hyperoperations or binary relations This ispossible because our considerations happen within a ring (of functions)

For an arbitrary pair of operators L(~p) L(~q) isin LAn(J)m where ~p = (pnminus1 p0)~q = (qnminus1 q0) we define an operation ldquomrdquo with respect to the m-th component byL(~p) m L(~q) = L(~u) where ~u = (unminus1 u0) and

uk(x) = pm(x)qk(x) + (1minus δkm)pk(x) (5)

for all k = nminus 1 0 k 6= m and all x isin J Obviously such an operation is not commuta-tive

Moreover apart from the above binary operation we can define also a relation ldquolemrdquocomparing the operators by their m-th component putting L(~p) lem L(~q) whenever for allx isin J there is

pm(x) = qm(x) and at the same time pk(x) le qk(x) (6)

for all k = nminus 1 0 Obviously (LAn(J)mlem) is a partially ordered setAt this stage in order to simplify the notation we write LAn(J) instead of LAn(J)m

because the lower index m is kept in the operation and relation The following lemma isproved in [2]

Lemma 1 Triads (LAn(J) mlem) are partially ordered (noncommutative) groups

Now we can use Lemma 1 to construct a (noncommutative) hypergroup In orderto do this we will need the following lemma known as Ends lemma for details seeeg [18ndash20] Notice that a join space is a special case of a hypergroupndashin this paper wespeak of hypergroups because we want to stress the parallel with groups

Lemma 2 Let (H middotle) be a partially ordered semigroup Then (H lowast) where lowast H times H rarr H isdefined for all a b isin H by

a lowast b = [a middot b)le = x isin H | a middot b le x

is a semihypergroup which is commutative if and only if ldquomiddotrdquo is commutative Moreover if (H middot) isa group then (H lowast) is a hypergroup

Thus to be more precise defining

m LAn(J)timesLAn(J)rarr P(LAn(J)) (7)

byL(~p) m L(~q) = L(~u) | L(~p) m L(~q) lem L(~u) (8)

for all pairs L(~p) L(~q) isin LAn(J)m lets us state the following lemma

Lemma 3 Triads (LAn(J) m) are (noncommutative) hypergroups

117


Notation 1 Hypergroups (LAn(J) m) will be denoted by HLAn(J)m for an easier distinction

Remark 1 As a parallel to (2) and (3) we define

L(qn q0)y(x) =n

sumk=0

qk(x)y(k)(x) q0 6= 0 qk isin C(J) (9)

andLAn(J) = qn q0) | q0 6= 0 qk(x) isin C(J) (10)

and by defining the binary operation ldquomrdquo and ldquolemrdquo in the same way as for LAn(J)m it is easy toverify that also (LAn(J) mlem) are noncommutative partially ordered groups Moreover given ahyperoperation defined in a way parallel to (8) we obtain hypergroups (LAn(J)m m) which willbe in line with Notation 1 denoted HLAn(J)m

22 Results

In this subsection we will construct certain mappings between groups or hypergroupsof linear differential operators of various orders The result will have a form of sequencesof groups or hypergroups

Define mappings Fn LAn(J)rarr LAnminus1(J) by

Fn(L(pnminus1 p0)) = L(pnminus2 p0)

and φn LAn(J)rarr LAnminus1(J) by

φn (L(pnminus1 p0)) = L(pnminus2 p0)

It can be easily verify that both Fn and φn are for an arbitrary n ge 2 group homomor-phisms

Evidently LAn(J) sub LAn(J)LAnminus1(J) sub LAn(J) for all admissible n isin N Thuswe obtain two complete sequences of ordinary linear differential operators with linkinghomomorphisms Fn and φn

LA0(J)id01 LA1(J)

id12 LA2(J)id23

LA0(J)

id0

OO

LA1(J)

id1

OO

F1oo

φ1

ii

LA2(J)

id2

OO

F2oo

φ2hh

F3oo

φ3gg

LAnminus2(J)idnminus2nminus1 LAnminus1(J)

idnminus1n LAn(J)idnn+1

LAnminus2(J)

idnminus2

OO

LAnminus1(J)

idnminus1

OO

Fnminus1oo

φnminus2ii

LAn(J)

idn

OO

Fnoo

φnhh

Fn+1oo

φn+1gg

(11)

where idkk+1 idk are corresponding inclusion embeddingsNotice that this diagram presented at the level of groups can be lifted to the level

of hypergroups In order to do this one can use Lemma 3 and Remark 1 However thisis not enough Yet as Lemma 4 suggests it is possible to show that the below presentedassignment is functorial ie not only objects are mapped onto objects but also morphisms(isotone group homomorphisms) are mapped onto morphisms (hypergroup homomor-phisms) Notice that Lemma 4 was originally proved in [4] However given the minimalimpact of the proceedings and its very limited availability and accessibility we include ithere with a complete proof

118


Lemma 4 Let (Gk middotklek) k = 1 2 be preordered groups and f (G1 middot1le1) rarr (G2 middot2le2)a group homomorphism which is isotone ie the mapping f (G1le1) rarr (G2le2) is order-preserving Let (Hk lowastk) k = 1 2 be hypergroups constructed from (Gk middotklek) k = 1 2 byLemma 2 respectively Then f (H1 lowast1) rarr (H2 lowast2) is a homomorphism ie f (a lowast1 b) subef (a) lowast2 f (b) for any pair of elements a b isin H1

Proof Let a b isin H1 be a pair of elements and c isin f (a lowast1 b) be an arbitrary element Thenthere is d isin a lowast1 b = [a middot1 b)le1 ie a middot1 b le1 d such that c = f (d) Since the mappingf is an isotone homomorphism we have f (a) middot2 f (b) = f (a middot1 b) le f (d) = c thus c isin[ f (a) middot2 f (b))le2 Hence

f (a lowast1 b) = f ([a middot1 b)le1) sube [ f (a) middot2 f (b))le = f (a) lowast2 f (b)

Consider a sequence of partially ordered groups of linear differential operators

LA0(J)F1larrminus LA1(J)

F2larrminus LA2(J)F3larrminus

Fnminus2larrminusminus LAnminus2(J)

Fnminus1larrminusminus LAnminus1(J) Fnlarrminus LAn(J)Fn+1larrminusminus LAn+1(J)larr

given above with their linking group homomorphisms Fk LAk(J) rarr LAkminus1(J) fork = 1 2 Since mappings Fn LAn(J)rarr LAnminus1(J) or rather

Fn (LAn(J) mlem)rarr (LAnminus1(J) mlem)

for all n ge 2 are group homomorphisms and obviously mappings isotone with respect tothe corresponding orderings we immediately get the following theorem

Theorem 1 Suppose J sube R is an open interval n isin N is an integer n = 2 m isin N suchthat m 5 n Let (HLAn(J)m lowastm) be the hypergroup obtained from the group (LAn(J)m m) byLemma 2 Suppose that Fn (LAn(J)m m)rarr (LAnminus1(J)m m) are the above defined surjectivegroup-homomorphisms n isin N n = 2 Then Fn (HLAn(J)m lowastm) rarr HLAnminus1(J)m lowastm) aresurjective homomorphisms of hypergroups

Proof See the reasoning preceding the theorem

Remark 2 It is easy to see that the second sequence from (11) can be mapped onto the sequenceof hypergroups

HLA0(J)mF1larrminus HLA1(J)m

F2larrminus HLA2(J)mF3larrminus

Fnminus2larrminusminus HLAnminus1(J)m

Fnminus1larrminusminus HLAn(J)m larr

This mapping is bijective and the linking mappings are surjective homomorphisms Fn Thusthis mapping is functorial

3 Automata and Related Concepts31 Notation and Preliminaries

The concept of an automaton is mathematical interpretation of diverse real-life systemsthat work on a discrete time-scale Various types of automata called also machines areapplied and used in numerous forms such as money changing devices various calculatingmachines computers telephone switch boards selectors or lift switchings and othertechnical objects All the above mentioned devices have one aspect in commonndashstates areswitched from one to another based on outside influences (such as electrical or mechanicalimpulses) called inputs Using the binary operation of concatenation of chains of input

119


symbols one obtains automata with input alphabets in the form of semigroups or a groupsIn the case of our paper we work with input sets in the form of hypercompositionalstructures When focusing on the structure given by transition function and simultaneouslyneglecting the output functions and output sets one reaches a generalization of automatandashquasi-automata (or semiautomata) see classical works such as eg [31821ndash24]

To be more precise a quasi-automaton is a system (A S δ) which consists of a non-void set A an arbitrary semigroup S and a mapping δ Atimes Srarr A such that

δ(δ(a r s)) = δ(a r s) (12)

for arbitrary a isin A and r s isin S Notice that the concept of quasi-automaton has beenintroduced by S Ginsberg as quasi-machine and was meant to be a generalization of theMealy-type automaton Condition (12) is sometimes called Mixed Associativity Condition(MAC) With most authors it is nameless though

For further reading on automata theory and its links to the theory of hypercomposi-tional structures (also known as algebraic hyperstructures) see eg [24ndash26] Furthermorefor clarification and evolution of terminology see [8] For results obtained by means ofquasi-multiautomata see eg [5ndash827]

Definition 1 Let A be a nonempty set (H middot) be a semihypergroup and δ A times H rarr A amapping satisfying the condition

δ(δ(s a) b) isin δ(s a middot b) (13)

for any triad (s a b) isin A times H times H where δ(s a middot b) = δ(s x) x isin a middot b Then the triad(A H δ) is called quasi-multiautomaton with the state set A and the input semihypergroups (H middot)The mapping δ A times H rarr A is called the transition function (or the next-state function) ofthe quasi-multiautomaton (A H δ) Condition (13) is called Generalized Mixed AssociativityCondition (or GMAC)

In this section Rn[x] means as usually the ring of polynomials of degree at most n

32 Results

Now consider linear differential operators L(m pnminus1 p0) Cinfin(R) rarr Cinfin(R)defined by

L(m pnminus1 p0) f = mdn f (x)

dxn +nminus1

sumk=0

pk(x)dk f (x)

dxk (14)

Denote byLA1An(R) the additive abelian group of differential operators L(m pnminus1 p0) where for L(m pnminus1 p0) L(k qnminus1 q0) isin LA1An(R) we define

L(m pnminus1 p0) + L(k qnminus1 q0) = L(m + k pnminus1 + qnminus1 p0 + q0) (15)

where

L(m + k pnminus1 + qnminus1 p0 + q0) f = (m + k)dn f (x)

dxn +nminus1

sumk=0

(pk(x) + qk(x))dk f (x)

dxk (16)

Suppose that pk isin Rnminus1[x] and define

δn Rn[x]times LA1An(R)rarr Rn[x] (17)

by

δn( f L(m pnminus1 p0)) = mdn f (x)

dxn + f (x) + m +nminus1

sumk=0

pk(x) f isin Rn[x] (18)

120


Theorem 2 Let LA1An(R) Rn[x] be structures and δn Rn[x] times LA1An(R) rarr Rn[x] themapping defined above Then the triad (Rn[x]LA1An(R) δn) is a quasi-automaton ie an actionof the group LA1An(R) on the group Rn[x]

Proof We are going to verify the mixed associativity condition (MAC) which shouldsatisfy the above defined action

Suppose f isin Rn[x] f (x) = sumnk=0 akxk L(m pnminus1 p0) L(k qnminus1 q0)

isin LA1An(R) Then

δn(δn( f L(m pnminus1 p0)) L(k qnminus1 q0)) =

= δn

(m

dn f (x)dxn + f (x) + m +

nminus1

sumk=0

pk(x) L(k qnminus1 q0)

)=

= δn

(m middot n middot an + m + f (x) +

nminus1

sumk=0


)=

= kdn f (x)

dxn + m middot n middot an + m + f (x) +nminus1

sumk=0

pk(x) +nminus1

sumk=0

qk(x) + k =

= (m + k)n middot an + (m + k) + f (x) +nminus1

sumk=0

(pk(x) + qk(x)) =

= (m + k)(n middot an + 1) + f (x) +nminus1

sumk=0

(pk(x) + qk(x)) =

= (m + k)dn f (x)

dxn + f (x) + (m + k) +nminus1

sumk=0

(pk(x) + qk(x)) =

= δn( f L(m + k pnminus1 + qnminus1 p0 + q0)) =

= δn( f L(m pnminus1 p0) + L(k qnminus1 q0)) (19)

thus the mixed associativity condition is satisfied

Since Rn[x]LA1An(R) are endowed with naturally defined orderings Lemma 2 canbe straightforwardly applied to construct semihypergroups from them

Indeed for a pair of polynomials f g isin Rn[x] we put f le g whenever f (x) leg(x) z isin Rn[x] In such a case (Rn[x]le) is a partially ordered abelian group Now wedefine a binary hyperoperation

Rn[x]timesRn[x]rarr P(Rn[x]) (20)

by

f g = h h isin Rn[x] f (x) + g(x) le h(x) x isin R = [ f + g)le (21)

By Lemma 2 we have that (Rn[x] ) is a hypergroupMoreover defining

LA1An(R)timesLA1An(R)rarr P(LA1An(R)) (22)

by L(mminusminusrarrp(x))L(k

minusminusrarrq(x)) =

[L(m

minusminusrarrp(x)) + L(k

minusminusrarrq(x))

)le

=[

L(m + kminusminusrarrp(x) +

minusminusrarrq(x)

)le

=

L(rminusminusrarru(x)) m + k le r

minusminusrarrp(x) +

minusminusrarrq(x) le minusminusrarru(x) which means

pj(x) + qj(x) le uj(x)

where j = 0 1 nminus 1 we obtain again by Lemma 2 that the hypergroupoid (LA1An(R) )is a commutative semihypergroup

121


Finally define a mapping

σn LA1An(R)timesRn[x]rarr Rn[x] (23)

byσn(L(m pnminus1 p0 f )) = L(m p0 f + pnminus1 p0 f + p1 p0) (24)

Below in the proof of Theorem 3 we show that the mapping satisfies the GMAC con-dition

This allows us to construct a quasi-multiautomaton

Theorem 3 Suppose (LA1An(R) ) (Rn[x] ) are hypergroups constructed above and σn LA1An(R)timesRn[x]rarr Rn[x] is the above defined mapping Then the structure

((LA1An(R) ) (Rn[x] ) σn)

is a quasi-multiautomaton

Proof Suppose L(m~p) isin LA1An(R) f g isin Rn[x] Then

σn(σn(L(m~p) f ) g) = σn(L(m p f + pnminus1 p f + p1 p0) g) =

= L(m p g + p f + pnminus1 p g + p f + p1 p0) =

= L(m p (g + f ) + pnminus1 p (g + f ) + p1 p0) isinisin σn(L(m p h + pnminus1 p h + p1 p0) f g h isin Rn[x] f + g le h == σn(L m pnminus1 p1 p0) [ f + g)le) = σn(L(m~p) f g) (25)

hence the GMAC condition is satisfied

Now let us discuss actions on objects of different dimensions Recall that a homo-morphism of automaton (S G δS) into the automaton (T H δT) is a mapping F = φtimes ψ Stimes G rarr T times H such that φ S rarr T is a mapping and ψ G rarr H is a homomorphism(of semigroups or groups) such that for any pair [s g] isin Stimes G we have

φ(δS(s g)) = δT(φ(s) ψ(g)) ie φ δS = δT (φtimes ψ) (26)

In order to define homomorphisms of our considered actions and especially in order toconstruct a sequence of quasi-automata with decreasing dimensions of the correspondingobjects we need a different construction of a quasiautomaton

If f isin Rn[x] f (x) = sumnk=0 anxk and L(m~p) isin LA1An(R) we define

τn(L(m pnminus1 p0) f ) = L(m an + pnminus1 a1 + p0 + a0) (27)

Now if g isin Rn[x] g(x) = sumnk=0 bkxk we have

τn(τn(L(m pnminus1 p0) f ) g) =

= τn(L(m an + pnminus1 a1 + p1 + a0) g) =

= L(m an + bn + pnminus1 a1 + b1 + p0 + a0 + b0) =

= τn

(L(m pnminus1 p0)

n

sumk=0

(ak + bk)x

)=

= τn(L(m pnminus1 p0) f + g) (28)

Hence τn LA1An(R) times Rn[x] rarr LA1An(R) is the transition function (satisfyingMAC) of the automaton A = (LA1An(R)Rnminus1[x] τn)

122


Consider now two automatandashAnminus1 = (LA1Anminus1(R)Rnminus1[x] τnminus1) and the aboveone Define mappings

φn LA1An(R)rarr LA1Anminus1(R) ψn Rn[x]rarr Rnminus1[x] (29)

in the following way For L(m pnminus1 p0) isin LA1An(R) put

φn(L(m pnminus1 p0)) = L(m pnminus2 p0) isin LA1Anminus1(R) (30)

and for f isin Rn[x] f (x) = sumnk=0 akxk define

ψn( f ) = ψn

(n

sumk=0

akxk

)=

nminus1

sumk=0

akxk isin Rnminus1[x] (31)

Evidently there is ψn( f + g) = ψn( f ) + ψ(g) for any pair of polynomials f g isin Rn[x]

Theorem 4 Let φn LA1An(R) rarr LA1Anminus1(R) ψn Rn[x] rarr Rnminus1[x] τn LA1An(R)timesRn[x] rarr LA1An(R) n isin N n = 2 be mappings defined above Define Fn An rarr Anminus1as mapping

Fn = φn times ψn LA1An(R)timesRn[x]rarr LA1Anminus1(R)timesRnminus1[x]

Then the following diagram

LA1An(R)timesRn[x]τn

φntimesψn

LA1An(R)

φn

LA1Anminus1(R)timesRnminus1[x]

τnminus1 LA1Anminus1(R)

(32)

is commutative thus the mapping Fn = φn times ψn is a homomorphism of the automaton An =(LA1An(R)Rn[x] τn) into the automaton Anminus1 = (LA1Anminus1(R)Rnminus1[x] τnminus1)

Proof Let [L(m~p) f ] isin LA1An(R)timesRn[z] f (x) = sumnk=0 akxk Then

(φn τn)(L(m~p) f ) = φn

(τn

(L(m pnminus1 p0)

n

sumk=0

akxk

))=

= φn((m an + pnminus1 a1 + p0 + a0)) = L(m anminus1 + pnminus2 a1 + p0 + a0) =

= τnminus1

(L(m pnminus2 p0)

nminus1

sumk=0

akxk

)=

= τnminus1

((φn times ψn)

(L(m pnminus1 p0)

n

sumk=0

akxk

))=

= (τnminus1 (φn times φn)))(L(m~p) f ) (33)

Thus the diagram (32) is commutative

Using the above defined homomorphism of automata we obtain the sequence ofautomata with linking homomorphisms Fk Ak rarr Anminus1 k isin N k ge 2

Fnminus1larrminusminus (LA1Anminus1(R)Rnminus1[x] τnminus1)

Fnlarrminus (LA1An(R)Rn[x] τn)

(LA1A1(R) τ1)F2larrminus (LA1A2(R) τ2)

F3larrminus Fnminus2larrminusminus (LA1Anminus2(R)Rnminus2[x] τnminus2)

Fnminus1larrminusminus (34)

123


Here for L(m Pminus 1 Pminus 0) isin LA1A2(R) we have

L(m p1 p0)y(x) = md2y(x)

dx2 + p1(x)dy(x)

dx+ p0(x)y(x) (35)

for any y(x) isin C2(R) and any L(m p0) isin LA1A1(R) it holds L(m p0)y(x) = m dy(x)dx +

p0(x)y(x)The obtained sequence of automata can be transformed into a countable sequence of

quasi-multiautomata We already know that the transition function

σn LA1An(R)timesRn[x]rarr LA1An(R)

satisfies GMAC Further suppose f g isin Rn[x] f (x) = summk=0 akxk g(x) = summ

k=0 bkxk Then

ψn( f g) = ψn(h h isin Rn[x] f + g 5 h) == ψn(h) h isin Rn[x] f + g 5 h == u u isin Rnminus1[x] ψn( f ) + ψn(g) 5 u =

= ψn( f )ψn(g)

here grad ψn(h) lt grad h for any polynomial h isin f g Thus the mapping ψn (Rn[x] )rarr(Rnminus1[x] ) is a good homomorphism of corresponding hypergroups

Now we are going to construct a sequence of automata with increasing dimensionsie in a certain sense sequence ldquodualrdquo to the previous sequence First of all we need acertain ldquoreductionrdquo member to the definition of a transition function

λlowastn LA1An(R)timesRn[x]rarr LA1An(R) (36)

namely rednminus1 Rr[x] rarr Rnminus1[x] whenever r gt nminus 1 In detail if f (x) = sumrkminus0 akxk isin

Rr[x] then

rednminus1( f ) = rednminus1

(r

sumk=0

akxk

)= rednminus1

(r

sumk=n

akxk +nminus1

sumk=0

akxkminus1

)=

=nminus1

sumk=0

akxkminus1 isin Rnminus1[x] (37)

Further L(m pnminus2 p0)n isin LA1An(R) is acting by

L(m pnminus2 p0)ny(x) = mdny(x)

dxn +nminus2

sumk=0

pk(x)dky(x)

dxk (38)

whereas L(m pnminus2 p0)nminus1 isin LA1Anminus1(R) ie

L(m pnminus2 p0)nminus1y(x) = mdnminus1y(x)

dxnminus1 +nminus2

sumk=0

pk(x)dky(x)

dxk (39)

Then for any pair (L(m pnminus1 p0) f ) isin LA1An(R)timesRn[x] where f (x) = sumnk=0 akxk

we obtain

λn = λn (idtimes rednminus1) (40)

124


and

(λn (idtimes rednminus1))(L(m pnminus1 p0 f ) = λ

n

(L(m pnminus1 p0)

nminus1

sumk=0

akxk

)=

= L(m pnminus1 + anminus1 p0 + a0) isin LA1An(R) (41)

thus the mapping λn LA1An(R)timesRn[x]rarr LA1An(R) is well defined We should verifyvalidity of MAC and commutativity of the square diagram determining a homomorphismbetween automata

Suppose L(m pnminus1 p0) isin LA1An(R) f g isin Rn[x] f (x) = sumnk=0 akxk g(x) =

sumnk=0 bkxk Then

λn(λn(L(m pnminus1 p0) f ) g) =

= λn(L(m pnminus1 ++anminus1 p0 + a0 g) =

= L(m pnminus1 + anminus1 + bnminus1 p0 + a0 + b0) =

= λn

(L(m pnminus1 p0)

n

sumk=0

(ak + bk)xk

)=

= λn(L(m pnminus1 p0) f + g) (42)

thus MAC is satisfiedFurther we are going to verify commutativity of this diagram

LA1Anminus1(R)timesRnminus1λnminus1

ξnminus1timesηnminus1

LA1Anminus1(R)

ηnminus1

LA1An(R)timesRn[x]

λn LA1An(R)

(43)

where ξnminus1(L(m pnminus2 p0)nminus1) = L(m pnminus1 p0)n ie

mdnminus1

dxnminus1 + pnminus1(x)dnminus2

dxnminus1 + middot middot middot+ p0(x)idrarr mdn

dxn + pnminus2(x)dnminus2

dxnminus2 + middot middot middot+ p0(x)id (44)

and ηnminus1

(sumnminus1

k=0 akxk)= anminus1xn + sumnminus1

k=0 akxk

Considering L(m pnminus2 p0) and the polynomial f (x) = sumnminus1k=0 akxk similarly as

above we have

(ηnminus1)(L(m pnminus2 p0) f ) = ηnminus1(L(m pnminus2 + anminus2 p0 + a0)nminus1) =

= ηnminus1(λnminus1(L(m pnminus2 p0) f ) = (ηnminus1 λnminus1)(L(m pnminus2 p0) f ) (45)

Thus the above diagram is commutative Now denoting by Tnminus1 the pair of mappings(ξnminus1 ηnminus1) we obtain that Tnminus1 (LA1Anminus1(R)Rnminus1[x] λnminus1)rarr (LA1An(R)Rn[x] λn)is a homomorphism of the given automata Finally using Tk as connecting homomorphismwe obtain the sequence

(LA1A1(R)R1[x] λ1)T1minusrarr (LA1A2(R)R2[x] λ2)

T2minusrarr (LA1A3(R)R3[x] λ3)T3minusrarr

Tnminus2minusminusrarr (LA1Anminus1(R)Rnminus1[x] λnminus1)

Tnminus1minusminusrarr (LA1An(R)Rn[x] λn)Tnminusrarr (46)

4 Practical Applications of the Sequences

In this section we will include several examples of the above reasoning We willapply the theoretical results in the area of artificial neurons ie in a way continue withthe paper [11] which focuses on artificial neurons For notation recall [11] Further on weconsider a generalization of the usual concept of artificial neurons We assume that the

125


inputs uxi and weight wi are functions of an argument t which belongs into a linearlyordered (tempus) set T with the least element 0 The index set is in our case the set C(J) ofall continuous functions defined on an open interval J sub R Now denote by W the set ofall non-negative functions w T rarr R Obviously W is a subsemiring of the ring of all realfunctions of one real variable x R rarr R Further denote by Ne(~wr) = Ne(wr1 wrn)for r isin C(J) n isin N the mapping

yr(t) =n

sumk=1

wrk(t)xrk(t) + br

which will be called the artificial neuron with the bias br isin R By AN(T) we denote thecollection of all such artificial neurons

41 Cascades of Neurons Determined by Right Translations

Similarly as in the group of linear differential operators we will define a binary opera-tion in the systemAN(T) of artificial neurons Ne(middot) and construct a non-commutative group

Suppose Ne(~wr) Ne(~ws) isin AN(T) such that r s isin C(J) and ~wr = (wr1 wrn)~ws = (ws1 wsn) where n isin N Let m isin N 1 le m le n be a such an integer thatwrm gt 0 We define

Ne(~wr) middotm Ne(~ws) = Ne(~wu)

where~wu = (wu1 wun) = (wu1(t) wun(t))

~wuk(t) = wrm(t)wsk(t) + (1minus δmk)wrk(t) t isin T

and of course the neuron Ne(~wu) is defined as the mapping yu(t) =nsum

k=1wk(t)xk(t) +

bu t isin T bu = brbsThe algebraic structure (AN(T) middotm) is a non-commutative group We proceed to

the construction of the cascade of neurons Let (Z+) be the additive group of all inte-gers Let Ne(~ws(t)) isin AN(T) be an arbitrary but fixed chosen artificial neuron with theoutput function

ys(t) =n

sumk=1

wsk(t)xsk(t) + bs

Denote by ρs AN(T)rarr AN(T) the right translation within the group of time varyingneurons determined by Ne(~ws(t)) ie

ρs(Ne(~wp(t)) = Ne(~wp(t) middotm Ne(~ws(t))

for any neuron Ne(~wp(t)) isin AN(T) In what follows denote by ρrs the r-th iteration of ρs

for r isin Z Define the projection πs AN(T)timesZrarr AN(T) by

πs(Ne(~wp(t)) r) = ρrs(Ne(~wp(t))

One easily observes that we get a usual (discrete) transformation group ie the actionof (Z+) (as the phase group) on the group AN(T) Thus the following two requirementsare satisfied

1 πs(Ne(~wp(t)) 0) = Ne(~wp(t)) for any neuron Ne(~wp(t)) isin AN(T)2 πs(Ne(~wp(t)) r + u) = πs(πs(Ne(~wp(t)) r) u) for any integers r u isin Z and any

artificial neuron Ne(~wp(t)) Notice that the just obtained structure is called a cascadewithin the framework of the dynamical system theory

42 An Additive Group of Differential Neurons

As usually denote by Rn[t] the ring of polynomials of variable t over R of the grade atmost n isin N0 Suppose ~w = (w1(t) wn(t)) be the fixed vector of continuous functions

126


wk R rarr R bp be the bias for any polynomial p isin Rn[t] For any such polynomialp isin Rn[t] we define a differential neuron DNe(~w) given by the action

y(t) =n

sumk=1

wk(t)dkminus1 p(t)

dtkminus1 + b0dn p(t)

dtn (47)

Considering the additive group of Rn[t] we obtain an additive commutative groupDN(T)of differential neurons which is assigned to the group of Rn[t] Thus for DNep(~w)DNeq(~w) isin DN(T) with actions

y(t) =n

sumk=1

wk(t)dkminus1 p(t)


dtn

and

z(t) =n

sumk=1

wk(t)dkminus1q(t)

dtkminus1 + b0dnq(t)

dtn

we have DNep+q(~w) = DNep(~w) + DNeq(~w) isin DN(T) with the action

u(t) = y(t) + z(t) =n

sumk=1

wk(t)dkminus1(p(t) + q(t))

dtkminus1 + b0dn(p(t) + q(t))

dtn

Considering the chain of inclusions

Rn[t] sub Rn+1[t] sub Rn+2[t]

we obtain the corresponding sequence of commutative groups of differential neurons

43 A Cyclic Subgroup of the Group AN(T)m Generated by Neuron Ne(~wr) isin AN(T)m

First of all recall that if Ne(~wr) Ne(~ws) isin AN(T)m r s isin C(J) where~wr(t) = (wr1(t) wrn(t)) ~ws(t) = (ws1(t) wsn(t)) are vector function of weights

such that wrm(t) 6= 0 6= wsm(t) t isin T with outputs yr(t) =nsum

k=1wrk(t)xk(t) + br ys(t) =

nsum

k=1wsk(t)xk(t) + bs (with inputs xk(t)) then the product Ne(~wr) middotm Ne(~ws) = Ne(~wu) has

the vector of weights~wu(t) = (wu1(t) wun(t))

with wuk(t) = wrm(t)wsk(t) + (1minus δmk)wrk(t) t isin T

The binary operation ldquomiddotmrdquo is defined under the assumption that all values of functionswhich are m-th components of corresponding vectors of weights are different from zero

Let us denote by ZANr(T) the cyclic subgroup of the group AN(T)m generated bythe neuron Ne(~wr) isin AN(T)m Then denoting the neutral element by N1(~e)m we haveZANr(T) =

= [Ne(~wr)]minus2 [Ne(~wr)]

minus1 N1(~e)m Ne(~wr) [Ne(~wr)]2 [Ne(~wr)]

p

Now we describe in detail objects

[Ne(~wr)]2 [Ne(~wr)]

p p isin N p ge 2 N1(~e)m and [Ne(~wr)]minus1 (48)

ie the inverse element to the neuron Ne(~wr)Let us denote [Ne(~wr)]2 = Ne(~ws) with ~ws(t) = (ws1(t) wsn(t)) then

wsk(t) = wrm(t)wrk(t) + (1minus δmk)wrk(t) = (wrm(t) + 1minus δmk)wrk(t)

127


Then the vector of weights of the neuron [Ne(~wr)]2 is

~ws(t) = ((wrm(t) + 1)wr1(t) w2rm(t) (wrm(t) + 1)wrn(t))

the output function is of the form

ys(t) =n

sumk=1k 6=m

(wrm(t) + 1)wrk(t)xk(t)) + w2rmxn(t) + b2

r

It is easy to calculate the vector of weights of the neuron [Ne(~wr)]3

((w2rm(t) + 1)wr1(t) w3

rm(t) (w2rm(t) + 1)wrn(t))

Finally putting [Ne(~wr)]p = Ne(~wv) for p isin N p ge 2 the vector of weights of thisneuron is

~wv(t) = ((wpminus1rm (t) + 1)wr1(t) wp

rm(t) (wpminus1rm (t) + 1)wrn(t))

Now consider the neutral element (the unit) N1(~e)m of the cyclic group ZANr(T)Here the vector~e of weights is~e = (e1 em en) where em = 1 and ek = 0 for eachk 6= m Moreover the bias b = 1

We calculate products Ne(~ws) middot N1(~e)m N1(~e)m) middot Ne(~ws) Denote Ne(~wu) Ne(~wv)results of corresponding products respectivelyndashwe have ~wu(t) = (wu1(t) wun(t))where

wuk(t) = wsm(t)ek(t) + (1minus δmk)wsk(t) = wsk(t)

if k 6= m and wuk(t) = wsm(t)(em(t) + 0 middot wsm(t)) = wsm(t) for k = m Since the biasis b = 1 we obtain yu(t) = xm(t) + 1 Thus Ne(~wu) = Ne(~ws) Similarly denoting~wv(t) = (wv1(t) wvn(t)) we obtain wvk(t) = em(t)wsk(t) + (1minus δmk)ek(t) = wsk(t)for k 6= m and wvk(t) = wsm(t) if k = m thus ~wv(t) = (ws1(t) wsn(t)) consequentlyNe(~wu) = Ne(~ws) again

Consider the inverse element [Ne(~wr)]minus1 to the element Ne(~wr isin ZAN(T)m DenoteNe(~ws) = [Ne(~wr)]minus1 ~ws(t) = (ws1(t) wsn(t)) t isin T We have Ne(~wr) middot Ne(~ws) =Ne(~wr) middotm [Ne(~wr)]minus1 = N1(~e)m Then

0 = e1 = wrm(t)ws1(t) + wr1(t)

0 = e2 = wrm(t)ws2(t) + wr2(t)

1 = em = wrm(t)wsm(t)

0 = en = wrm(t)wsn(t) + wrn(t)

From the above equalities we obtain

ws1(t) = minuswr1(t)wrm(t)

wsm(t) =1

wrm(t) wsn(t) = minus

wrn(t)wrm(t)

Hence for [Ne(~wr)]minus1 = Ne(~ws) we get

~ws(t) =(minus wr1(t)

wrm(t)

1wrm(t)

minus wrn(t)wrm(t)

)=

=1

wrm(t)(minuswr1(t) 1 minuswrn(t))

128


where the number 1 is on the m-th position Of course the bias of the neuron [Ne(~wr)]minus1 isbminus1

r where br is the bias of the neuron Ne(~wr)

5 Conclusions

The scientific school of O Boruvka and F Neuman used in their study of ordinarydifferencial equations and their transformations [128ndash30] the algebraic approach withthe group theory as a main tool In our study we extended this existing theory withthe employment of hypercomposiional structuresmdashsemihypergroups and hypergroupsWe constructed hypergroups of ordinary linear differential operators and certain sequencesof such structures This served as a background to investigate systems of artificial neuronsand neural networks

Author Contributions Investigation JC MN BS writingmdashoriginal draft preparation JC MNDS writingmdashreview and editing MN BS supervision JC All authors have read and agreed tothe published version of the manuscript

Funding This research received no external funding





References1 Neuman F Global Properties of Linear Ordinary Differential Equations Academia Praha-Kluwer Academic Publishers Dordrecht

The Netherlands Boston MA USA London UK 19912 Chvalina J Chvalinovaacute L Modelling of join spaces by n-th order linear ordinary differential operators In Proceedings of

the Fourth International Conference APLIMAT 2005 Bratislava Slovakia 1ndash4 February 2005 Slovak University of TechnologyBratislava Slovakia 2005 pp 279ndash284

3 Chvalina J Moucka J Actions of join spaces of continuous functions on hypergroups of second-order linear differentialoperators In Proc 6th Math Workshop with International Attendance FCI Univ of Tech Brno 2007 [CD-ROM] Available onlinehttpsmathfcevutbrczpribylworkshop_2007prispevkyChvalinaMouckapdf (accessed on 4 February 2021)

4 Chvalina J Novaacutek M Laplace-type transformation of a centralizer semihypergroup of Volterra integral operators withtranslation kernel In XXIV International Colloquium on the Acquisition Process Management University of Defense Brno CzechRepublic 2006

5 Chvalina J Krehliacutek Š Novaacutek M Cartesian composition and the problem of generalising the MAC condition to quasi-multiautomata Analele Universitatii ldquoOvidius Constanta-Seria Matematica 2016 24 79ndash100 [CrossRef]

6 Krehliacutek Š Vyroubalovaacute J The Symmetry of Lower and Upper Approximations Determined by a Cyclic Hypergroup Applicablein Control Theory Symmetry 2020 12 54 [CrossRef]

7 Novaacutek M Krehliacutek Š ELndashhyperstructures revisited Soft Comput 2018 22 7269ndash7280 [CrossRef]8 Novaacutek M Krehliacutek Š Stanek D n-ary Cartesian composition of automata Soft Comput 2020 24 1837ndash18499 Cesarano C A Note on Bi-orthogonal Polynomials and Functions Fluids 2020 5 105 [CrossRef]10 Cesarano C Multi-dimensional Chebyshev polynomials A non-conventional approach Commun Appl Ind Math 2019 10 1ndash19

[CrossRef]11 Chvalina J Smetana B Series of Semihypergroups of Time-Varying Artificial Neurons and Related Hyperstructures Symmetry

2019 11 927 [CrossRef]12 Jan J Digital Signal Filtering Analysis and Restoration IEEE Publishing London UK 200013 Koudelka V Raida Z Tobola P Simple electromagnetic modeling of small airplanes neural network approach Radioengineering

2009 18 38ndash4114 Krenker A Bešter J Kos A Introduction to the artificial neural networks In Artificial Neural Networks Methodological Advances

and Biomedical Applications Suzuki K Ed InTech Rijeka Croatia 2011 pp 3ndash1815 Raida Z Lukeš Z Otevrel V Modeling broadband microwave structures by artificial neural networks Radioengineering 2004

13 3ndash1116 Srivastava N Hinton G Krizhevsky A Sutskever I Salakhutdinov R Dropout a simple way to prevent neural networks

from overfitting J Mach Learn Res 2014 15 1929ndash195817 Neuman F Global theory of ordinary linear homogeneous differential equations in the ral domain Aequationes Math 1987

34 1ndash22 [CrossRef]

129


18 Krehliacutek Š Novaacutek M From lattices to Hvndashmatrices An St Univ Ovidius Constanta 2016 24 209ndash222 [CrossRef]19 Novaacutek M On EL-semihypergroups Eur J Comb 2015 44 274ndash286 ISSN 0195-6698 [CrossRef]20 Novaacutek M Some basic properties of EL-hyperstructures Eur J Comb 2013 34 446ndash459 [CrossRef]21 Bavel Z The source as a tool in automata Inf Control 1971 18 140ndash155 [CrossRef]22 Doumlrfler W The cartesian composition of automata Math Syst Theory 1978 11 239ndash257 [CrossRef]23 Geacutecseg F Peaacutek I Algebraic Theory of Automata Akadeacutemia Kiadoacute Budapest Hungary 197224 Massouros GG Hypercompositional structures in the theory of languages and automata An St Univ AI Ccediluza Iasi Sect Inform

1994 III 65ndash7325 Massouros GG Mittas JD Languages Automata and Hypercompositional Structures In Proceedings of the 4th International

Congress on Algebraic Hyperstructures and Applications Xanthi Greece 27ndash30 June 199026 Massouros C Massouros G Hypercompositional Algebra Computer Science and Geometry Mathematics 2020 8 138 [CrossRef]27 Krehliacutek Š n-Ary Cartesian Composition of Multiautomata with Internal Link for Autonomous Control of Lane Shifting

Mathematics 2020 8 835 [CrossRef]28 Boruvka O Lineare Differentialtransformationen 2 Ordnung VEB Deutscher Verlager der Wissenschaften Berlin Germany 196729 Boruvka O Linear Differential Transformations of the Second Order The English University Press London UK 197130 Boruvka O Foundations of the Theory of Groupoids and Groups VEB Deutscher Verlager der Wissenschaften Berlin Germany 1974

130

mathematics

Article

On Factorizable Semihypergroups

Dariush Heidari 1 and Irina Cristea 21 Faculty of Science Mahallat Institute of Higher Education Mahallat 37811-51958 Iran

dheidarimahallatacir or dheidari82gmailcom2 Centre for Information Technologies and Applied Mathematics University of Nova Gorica

Vipavska cesta 13 5000 Nova Gorica Slovenia Correspondence irinacristeaungsi or irinacriyahoocouk Tel +386-0533-15-395

Received 29 May 2020 Accepted 17 June 2020 Published 1 July 2020

Abstract In this paper we define and study the concept of the factorizable semihypergroup iea semihypergroup that can be written as a hyperproduct of two proper sub-semihypergroupsWe consider some classes of semihypergroups such as regular semihypergroups hypergroupsregular hypergroups and polygroups and investigate their factorization property

Keywords semihypergroup factorization regular semihypergroup polygroup

1 Introduction

The decomposition property of a set appears as a key element in many topics in algebrabeing connected for example with equivalence relations (and called partition) factorizablesemigroups [12] factorization of groups [3] breakable semigroups [4] or recently introduced breakablesemihypergroups [5] A group G is called factorized if it can be written as a product of two subgroupsA and B This means that any element g in G has the form g = ab for some a isin A and b isin B [3] One ofthe most famous results about factorization of groups proven in 1955 by Ito [6] concerns the productof two abelian subgroups More precisely it is proven that any group G = AB written as the productof two abelian subgroups is metabelian A survey on some topics related to factorization of groupswas published in 2009 by Amberg and Kazarin [3] The problem of the factorization of groups has beenimmediately extended to the theory of semigroups We mention here the systematic studies of Tolo [2]Catino [1] or Tirasupa [7] A semigroup is called factorizable if it can be written as the product of twoproper subgroups [1] Besides the concept of left univocal factorization (A B) of a semigroup S = ABis defined by the supplementary condition for any a aprime isin A and b bprime isin B the equality ab = aprimebprime

implies that a = aprime (and similarly for the right univocal factorization) In the same article [1] afterconstructing all the semigroups that have a univocal factorization with factors isomorphic with a pairof prescribed semigroups the author presented necessary and sufficient conditions that a factorizationof a semigroup by right simple semigroups be univocal and characterized the semigroups with thisfactorization property

Motivated by the above-mentioned studies in this paper we introduce the notion of the factorizablesemihypergroup and we investigate the factorization property for special semihypergroups as regularsemihypergroups hypergroups regular hypergroups and polygroups Then we also present someproperties connected with the left (right) univocal factorization It is worth mentioning here thatthe problem of decomposition or factorization in hypercompositional algebra has been previouslystudied by the authors in [5] related with breakable semihypergroups or by Massouros in [89] In [8]the separation aspect was studied for hypergroups join spaces and convexity hypergroups whilein [9] it was proven that from the general decomposition theorems which are valid in hypergroupswell known decomposition theorems for convex sets were derived as corollaries like Kakutanirsquoslemma Stone and Hellyrsquos theorem etc

131


2 Preliminaries

For the basic concepts and terminology of semihypergroups or hypergroups the reader is refereedto the fundamental books [10ndash12] In the following we will recall those related to the identity elementinvertible element or zero element as well as the basic properties of regular semihypergroups andregular hypergroups or polygroups See also [1314] for more details regarding small polygroups

Let S be a semihypergroup ie S is a nonempty set endowed with a hyperoperation StimesS minusrarr Plowast(S) so a function from the Cartesian product Stimes S to the family of nonempty subsets ofS which is associative any three elements a b c isin S satisfies the property (a b) c = a (b c)where the left side member of the equality means the union of all hyperproducts u c with u isin a bA semihypergroup is a hypergroup if the reproduction axiom is verified as well for any a isin S there isa S = S = S a Sometimes especially in the case of semihypergroups when there is no risk ofconfusion the hyperoperation is omitted and the hyperproduct between two elements is simplydenoted by ab as we will do also throughout this paper Besides we denote the cardinality of a set Sby |S| An element a isin S is called left (right) scalar if |ax| = 1 (respectively |xa| = 1) for all x isin S If ais both a left and right scalar then it is called a two-sided scalar or simply a scalar An element e in asemihypergroup S is called the left (right) identity if x isin ex (respectively x isin xe) for all x isin S If e isboth a left and an identity then it is called a two-sided identity or simply an identity An element e ina semihypergroup S is called a left (right) scalar identity if ex = x (respectively xe = x) for allx isin S If e is both a left and right scalar identity then it is called a two-sided scalar identity or simply ascalar identity An element aprime isin S is called a left (right) inverse of the element a isin S if there exists a left(right) identity e isin S such that e isin aprimea (respectively e isin aaprime) If aprime is both a left and right inverse of athen it is called a two-sided inverse or simply an inverse of a An element zero in a semihypergroup Sis called a left (right) zero element if 0x = 0 (respectively x0 = 0) for all x isin S If zero is both aleft and a right zero then it is called a two-sided zero or simply a zero

A regular hypergroup is a hypergroup that has at least one identity and every element has atleast one inverse The regularity property for semihypergroups is not the same in the sense thata semihypergroup S is called regular if every element x of S is regular ie if there exists s isin Ssuch that x isin xsx Notice that any regular semigroup is a regular semihypergroup and moreoverbased on the reproduction axiom every hypergroup is a regular semihypergroup (but not necessarilya regular hypergroup)

Definition 1 ([12]) A polygroup is a system 〈P 1minus1 〉 where 1 isin Pminus1 is a unitary operation on P mapsPtimes P into the family of non-empty subsets of P and the following axioms hold for all x y z isin P

(P1) x (y z) = (x y) z(P2) 1 x = x = x 1(P3) x isin y z implies y isin x zminus1 and z isin yminus1 x

Theorem 1 Let S be a semihypergroup with the scalar identity element one such that every element x isin S hasa unique inverse denoted by xminus1 Then S is a polygroup if and only if

(Pprime3) forallx y isin S (xy)minus1 = yminus1xminus1

where Aminus1 = aminus1 | a isin A for A sube S

Proof We can restrict our attention to the ldquoonly ifrdquo part since the ldquoifrdquo part is obvious Let x y z isin Ssuch that x isin yz Then 1 isin xxminus1 sube y(zxminus1) and thus yminus1 isin zxminus1 Hence the property (Pprime3) impliesy isin xzminus1 Similarly z isin yminus1x Therefore S is a polygroup

3 Factorizable Semihypergroups

Based on the definition of factorizable semigroups in this section we first introduce and studythe main properties of factorizable semihypergroups

132


Definition 2 A semihypergroup S is said to be factorizable if there exist some proper sub-semihypergroups Aand B of S such that S = AB and |ab| = 1 for every a isin A and b isin B The pair (A B) is called a factorizationof S with factors A and B

It is clear that every factorizable semigroup can be considered as a factorizable semihypergroupIn the following we give some examples of factorizable proper semihypergroups (ie they arenot semigroups)

Example 1 Let S = 1 2 3 be a semihypergroup represented by the Cayley table

1 2 31 1 1 12 1 2 33 1 2 3 3 2 3

Then the pair (A B) where A = 1 2 and B = 2 3 is a factorization of S

Example 2 Let S = 1 2 3 4 5 be a semihypergroup with the following Cayley table

1 2 3 4 51 1 2 3 4 52 2 1 3 4 53 3 3 123 5 454 4 4 5 124 355 5 5 45 35 12345

Then the pair (A B) where A = 1 2 3 and B = 1 2 4 is a factorization of S

Lemma 1 If a semihypergroup S is factorizable as S = AB where A has a left identity and B has a rightidentity then S has a two-sided identity

Proof Suppose that e is a left identity of the sub-semihypergroup A and eprime is a right identity ofthe sub-semihypergroup B Since e eprime isin S we have e = ab and eprime = aprimebprime for some a aprime isin A andb bprime isin B Thus e = ab isin a(beprime) = (ab)eprime = eeprime thus e = eeprime Similarly one obtains eprime = eeprime Thereforee = eprime isin A cap B is a two-sided identity element of S

Similarly we can prove the following lemma

Lemma 2 If a semihypergroup S is factorizable as S = AB where A has a left scalar identity and B has a rightscalar identity then A B and S have a two-sided scalar identity

In the next theorem we present a sufficient condition such that a factorizable semihypergroup isa hypergroup

Theorem 2 If a semihypergroup S is factorizable as S = AB where A and B are hypergroups such thatA cap B 6= empty then S is a hypergroup

Proof Let S be a semihypergroup and (A B) a factorization of S where A and B are hypergroupssuch that e isin A cap B Then for every x isin S there exist a isin A and b isin B such that x = ab By thereproducibility property of A and B there exist aprime aprimeprime alowast isin A and bprime bprimeprime blowast isin B such that a isin aprimeaprimeprimeaprimeprime isin ealowast b isin bprimebprimeprime and e isin bprimeblowast Thus we can write

x = ab isin (aprimeaprimeprime)b sube aprime(ealowast)b sube aprime(bprimeblowast)alowastb = (aprimebprime)(blowastalowastb)

133


Hence x isin (aprimebprime)z = yz with z isin blowastalowastb and y = aprimebprime Similarly one proves that x isin ty with t isin Smeaning that the reproducibility property in S holds as well Therefore S is a hypergroup

The following example shows that the converse of the above theorem does not hold necessarily

Example 3 Let S = 1 2 3 4 5 6 be a hypergroup with the Cayley table

1 2 3 4 5 61 1 2 3 4 5 62 2 1 4 3 6 53 1 2 3 4 5 64 2 1 4 3 6 55 135 246 135 246 135 2466 246 135 246 125 246 135

Then the pair (A B) where A = 1 2 3 4 and B = 1 3 5 is a factorization of S but A is not ahypergroup while B is a hypergroup

Theorem 3 Let the semihypergroup S be factorizable as S = AB where A is a group If B is contained in thesemihypergroup class Ci (i = 1 2 3) then so S is where

C1 = the class of regular semihypergroupsC2 = the class of hypergroupsC3 = the class of regular hypergroups

Proof We denote by e the identity of the group A which is based on Lemma 1 a left scalar identity of S

(1) Assume that B is a regular semihypergroup and let x = ab be an arbitrary element in S = ABThen there exits bprime isin B such that b isin bbprimeb Thus x = ab isin abbprimeaminus1ab and thereby x isin xxprimexfor some xprime isin bprimeaminus1 This means that any element in S is regular so the semihypergroup S isregular as well

(2) Assume that B is a hypergroup and S = AB Then there exist a isin A and b isin B such that e = abThus aminus1 = eb = b isin A cap B So by Theorem 2 we conclude that S is a hypergroup

(3) Assume that B is a regular hypergroup Then by Lemma 1 the identity elements of S A andB are the same Let x = ab with a isin A and b isin B be an arbitrary element of S and y isin bminus1aminus1Then ya = bminus1 hence 1 isin bminus1b = yab so y = (ab)minus1 Thus (ab)minus1 = bminus1aminus1 Therefore S is aregular hypergroup

Definition 3 A factorization (A B) of a semihypergroup S is called left univocal (respectively right univocal)if the following condition holds

(foralla aprime isin A b bprime isin B) ab = aprimebprime =rArr a = aprime( respectively b = bprime)

A factorization (A B) of a semihypergroup S is called univocal if it is both left and right univocal

Example 4 Let S = 1 2 3 4 be a semihypergroup with the following Cayley table

1 2 3 41 1 2 3 42 2 1 2 4 3 43 3 4 1 3 2 44 4 3 4 2 4 1 2 3 4

134


Then the pair (A B) where A = 1 2 and B = 1 3 is a left univocal factorization of S


1 2 3 4 5 61 1 2 3 4 5 62 2 1 3 5 4 63 3 3 1 2 6 6 4 54 4 5 6 1 2 35 5 4 6 2 1 36 6 6 4 5 3 3 1 2

Then the pair (A B) where A = 1 4 and B = 1 2 3 is a left univocal factorization of S

Lemma 3 Let (A B) be a left univocal factorization of a semihypergroup S Then the following assertions hold

(i) If a = aprimeb for a aprime isin A and b isin B then a = aprime(ii) Let a isin A A1 sube A and B1 sube B be such that a isin A1B1 Then a isin A1

(iii) Let a isin A A1 sube A b isin B and B1 sube B be such that ab isin A1B1 Then a isin A1(iv) If a isin A1b with A1 sube A then a isin A1 and a = ab(v) For every a isin A there exists b isin B such that a = ab

Proof

(i) Suppose that a = aprimeb for a aprime isin A and b isin B Then ab isin aprime(bb) and thus ab = aprimebprime for bprime isin bbwhich implies that a = aprime

(ii) Let a isin A A1 sube A and B1 sube B be such that a isin A1B1 Then we have a = a1b1 with a1 isin A1 andb1 isin B1 Hence Assertion (i) implies that a = a1 isin A1

(iii) Let a isin A A1 sube A b isin B and B1 sube B be such that ab isin A1B1 Then we have ab = a1b1with a1 isin A1 and b1 isin B so a = a1 isin A1

(iv) This follows immediately from Assertion (i)(v) For every a isin A there exist aprime isin A and b isin B such that a = aprimeb Then Assertion (i) implies a = aprime

as required

Lemma 4 Let (A B) be a left univocal factorization of a semihypergroup S Then Acap B is a left zero semigroupformed by right scalar identities of A

Proof First we prove that the intersection Acap B is nonempty Let b be an element of B such that b = ebprimewith e isin A and bprime isin B Then for every a isin A it follows that ab isin (ae)bprime thus Lemma 3 (iii) impliesa isin ae Again by Lemma 3 (v) there exists bprimeprime isin B such that e = ebprimeprime while bprimeprime = alowastblowast with alowast isin A andblowast isin B Thus alowastbprimeprime isin (alowastalowast)blowast It follows that alowast isin alowastalowast by Lemma 3 (iii) Hence alowast isin alowaste = alowastebprimeprime soalowast = alowastbprimeprime by Lemma 3 (iv) Thereby alowast = alowastbprimeprime = (alowastalowast)blowast and thus alowast = alowastblowast = bprimeprime which showsthat A cap B 6= empty

Now let e be an arbitrary element in A cap B Then ee isin A cap B with |ee| = 1 and e(ee) = (ee)ethus e = ee because of the univocal factorization Moreover since ae = (ae)e it follows that a = aethus e is a scalar right identity of A Therefore A cap B is a left zero sub-semigroup of S formed by rightscalar identities of A

Remark 1 Note that the property ldquoto be a scalarrdquo is essential in Lemma 4 ie if (A B) is a left univocalfactorization of a semihypergroup S and e is a right identity but not a scalar one then e isin A cap B as we can seein the following example

135



1 2 3 41 1 1 2 1 12 1 1 2 1 13 1 2 3 4 34 1 2 3 4 4

Then the pair (A B) where A = 3 4 and B = 1 2 4 is a factorization of S We observe that theelement 3 is a right but not scalar identity of A which does not belong to A cap B

Theorem 4 Let (A B) be a left univocal factorization of a polygroup P Then the following assertions hold

(i) A and B are sub-polygroups of P(ii) A cap B = 1 where one is the identity of the polygroup P

Proof

(i) Let P be a polygroup with the identity one and (A B) be a left univocal factorization of PFirst we prove that both A and B contain one Since there exists a isin A such that 1 = aaminus1it follows that a isin a2aminus1 and hence a = aprimeaminus1 for some aprime isin a2 Thus a isin aaminus1 = 1 hence1 = a = aminus1 isin A cap B

Now for every a isin A there exist aprime isin A and bprime isin B such that aminus1 = aprimebprime Thus 1 isin aaminus1 = aaprimebprimeand hence 1 isin aaprime by Lemma 3 (ii) This means that A is a polygroup

Furthermore let b isin B Then bminus1 = aprimebprime with aprime isin A and bprime isin B Thus 1 isin bminus1b = aprimebprimeb soaprime = 1 by Lemma 3 (iv) Hence bminus1 = bprime isin B meaning that B is a polygroup

(ii) This follows from Assertion (i) and Lemma 4

Example 7 Let P = 1 2 3 4 5 6 be a polygroup with the following Cayley table

1 2 3 4 5 61 1 2 3 4 5 62 2 1 3 5 4 63 3 3 12 6 6 454 4 5 6 1 2 35 5 4 6 2 1 36 6 6 45 3 3 12

Then the pair (A B) where A = 1 2 3 and B = 1 4 is a factorization of P

Lemma 5 Let (A B) be a left univocal factorization of a polygroup P Then |P| = |A| middot |B|

Proof Let (A B) be a left univocal factorization of a polygroup P Since P = AB it is sufficient toprove that for every x isin P there exist unique a isin A and b isin B such that x = ab If x = ab = aprimebprime thenby definition we get a = aprime and hence ab = abprime Therefore b isin aaminus1bprime Thus there exists ardquo isin Asuch that b isin ardquobprime Hence ardquo isin bbprimeminus1 It follows that aprimeprime isin A cap B meaning that ardquo = 1 by Theorem 4We conclude that b = bprime

Proposition 1 The minimum order of a non-commutative left univocal factorizable polygroup is six

Proof First we show that every left univocal factorizable polygroup of order less than six iscommutative Let (A B) be a left univocal factorization of a polygroup P and |P| lt 6 Then Lemma 5

136


implies |P| = 4 and |A| = |B| = 2 Consider P = 1 a b c A = 1 a and B = 1 b Since A andB are sub-polygroups of P by Theorem 2 we have a = aminus1 and b = bminus1 and hence c = cminus1 Nowfor every x y isin P we have

xy = (xy)minus1 = yminus1xminus1 = yx

Therefore P is a commutative polygroup Now one can see that the symmetric group on theset 1 2 3 denoted S3 = () (1 2) (1 3) (2 3) (1 2 3) (1 3 2) as a polygroup has a left univocalfactorization S3 = () (1 2)() (1 2 3) (1 3 2) which completes the proof

4 Conclusions

Many properties from semigroup theory have been extended to semihypergroup theory showingtheir similarities but also differences and the factorization property discussed within this note is oneof them Following the classical paper of Catino [1] we defined a factorizable semihypergroup as asemihypergroup that can be written as a hyperproduct of its two proper sub-semihypergroups One ofthe main results presented here showed that if the semihypergroup S is factorized as S = AB withA a group and B a regular semihypergroup a hypergroup or a regular hypergroup then also S hasthe same algebraic hypercompositional structure as B Regarding the polygroups it was proven thatfor a left univocal factorizable polygroup P = AB both factors are sub-polygroups of P with theintersection containing only the identity of P Moreover we determined that the minimum order of anon-commutative left univocal factorizable polygroup is six

Author Contributions Conceptualization DH and IC funding acquisition IC investigation DH and ICmethodology DH and IC writing original draft DH writing review and editing IC All authors read andagreed to the published version of the manuscript

Funding The second author acknowledges the financial support from the Slovenian Research Agency (ResearchCore Funding No P1-0285)


References

1 Catino F Factorizable semigroups Semigroup Forum 1987 36 167ndash1742 Tolo K Factorizable semigroups Pac J Math 1969 31 523ndash5353 Amberg B Kazarin LS Factorizations of groups and related topics Sci China Ser A Math 2009 52 217ndash2304 Redei L Algebra I Pergamon Press Oxford UK 19675 Heidari D Cristea I Breakable semihypergroups Symmetry 2019 11 1006 Ito N Uber das Produkt von zwei abelschen Gruppen Math Z 1955 62 400ndash4017 Tirasupa Y Factorizable transformation semigroups Semigroup Forum 1979 18 15ndash198 Massouros CG Separation and Relevant Properties in Hypergroups AIP Conf Proc 2016 1738

480051-1ndash480051-49 Massouros CG On connections between vector spaces and hypercompositional structures Int J Pure

Appl Math 2015 34 133ndash15010 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Tricesimo Italy 199311 Davvaz B Semihypergroup Theory ElsevierAcademic Press London UK 201612 Davvaz B Polygroup Theory and Related Systems World Sciences Publishing Co Pte Ltd Hackensack NJ

USA 201313 Heidari D Davvaz B Characterization of small polygroups by their fundamental groups Discret Math

Algorithms Appl 2019 11 195004714 Heidari D Amooshahi M Davvaz B Generalized Cayley graphs over polygroups Commun Algebra 2019

47 2209ndash2219


137

mathematics

Article

1-Hypergroups of Small Sizes

Mario De Salvo 1 Dario Fasino 2 Domenico Freni 2 and Giovanni Lo Faro 1

Citation De Salvo M Fasino D

Freni D Lo Faro G 1-Hypergroups

of Small Sizes Mathematics 2021 9

108 httpsdoiorg103390

math9020108







nal affiliations








40)

1 Dipartimento di Scienze Matematiche e Informatiche Scienze Fisiche e Scienze della TerraUniversitagrave di Messina 98122 Messina Italy lofarounimeit

2 Dipartimento di Scienze Matematiche Informatiche e Fisiche Universitagrave di Udine 33100 Udine Italydariofasinouniudit (DF) domenicofreniuniudit (DF)

Correspondence desalvounimeit

Abstract In this paper we show a new construction of hypergroups that under appropriateconditions are complete hypergroups or non-complete 1-hypergroups Furthermore we classify the1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β Many ofthese hypergroups can be obtained using the aforesaid hypergroup construction

Keywords hypergroups complete hypergroups fundamental relations

1 Introduction

Hypercompositional algebra is a branch of Algebra experiencing a surge of activitynowadays that concerns the study of hyperstructures that is algebraic structures wherethe composition of two elements is a set rather than a single element [1] The subjectsmethods and goals of the hypercompositional algebra are very different from those ofclassic algebra However the two fields are connected by certain equivalence relationscalled fundamental relations [23] Through fundamental relations the analysis of algebraichyperstructures can make use of the wealth of tools typical of classical algebra Indeedfundamental relations are peculiar equivalence relations defined on hyperstructures insuch a way that the associated quotient set is one of the classical algebraic structures

More precisely a fundamental relation is the smallest equivalence relation definedon the support of a hyperstructure such that the corresponding quotient set is a classicalstructure having operational properties analogous to those of the hyperstructure [4ndash7] Forexample the quotient structure modulo the equivalence βlowast defined on a semihypergroup(or a hypergroup) is a semigroup (or a group respectively) [28ndash10] Analogous definitionsand results are also known in hyperstructures endowed with more than one operation seeeg [11] Moreover hypergroups can be classified according to the height of a βlowast-classthat is the least number of order-2 hyperproducts that can cover that class see [12]

If (H ) is a hypergroup and ϕ H 7rarr Hβlowast is the canonical projection then the kernelωH = ϕminus1(1Hβlowast) is the heart of (H ) The heart of a hypergroup plays a very importantrole in hypergroup theory Indeed if we know the structure of ωH then we have detailedinformation on the partition determined by relation βlowast since βlowast(x) = ωH x = x ωH forall x isin H When the heart of a hypergroup (H ) has only one element ε this element is alsothe identity of (H ) since x isin βlowast(x) = x ε = ε x According to a definition introducedby Corsini in [4] the hypergroups whose heart has size 1 are called 1-hypergroups In ([12]Theorem 2) we characterized the 1-hypergroups in terms of the height of their heartand in [13] Sadrabadi and Davvaz investigated sequences of join spaces associated withnon-complete 1-hypergroups

In this paper we deepen the knowledge of 1-hypergroups In particular we classifythe 1-hypergroups of cardinalities up to 6 on the basis of the partition of H induced byβlowast This technique allows us to explicitly construct all 1-hypergroups of order 5 andenumerate those of order 6 by means of scientific computing software We recall thatthe study of small-size algebraic hyperstructures is both a practical tool to analyze more

139


elaborate structures and a well-established research topic in itself In fact the enumerationand classification of hyperstructures having small cardinality have made it possible tosolve various relevant existence issues in hyperstructure theory see eg [14ndash17]

The plan of this paper is the following In the forthcoming Section 2 we introducethe basic definitions notations and fundamental facts to be used throughout the paperIn Section 3 we present a new construction of hypergroups that under appropriate hy-potheses are complete hypergroups or non-complete 1-hypergroups Moreover we provea few results concerning the β-classes of 1-hypergroups and sufficient conditions for 1-hypergroups to be complete which are relevant in subsequent sections In Section 4 wedetermine the 1-hypergroups of size 5 up to isomorphisms In Section 5 we classify the1-hypergroups of size 6 up to isomorphisms The 1-hypergroups of size 4 and many1-hypergroups of size 5 and 6 can be determined by the construction defined in Section 3The paper ends with some conclusions and directions for future research in Section 6

2 Basic Definitions and Results

Let H be a non-empty set and let Plowast(H) be the set of all non-empty subsets of H Ahyperproduct on H is a map from HtimesH to Plowast(H) For all x y isin H the subset x y is thehyperproduct of x and y If A B are non-empty subsets of H then A B =

⋃xisinAyisinB x y

A semihypergroup is a non-empty set H endowed with an associative hyperproduct that is (x y) z = x (y z) for all x y z isin H We say that a semihypergroup (H ) isa hypergroup if for all x isin H we have x H = H x = H the so-called reproducibilityproperty

A non-empty subset K of a semihypergroup (H ) is called a subsemihypergroup of(H ) if it is closed with respect to the hyperproduct that is x y sube K for all x y isin KA non-empty subset K of a hypergroup (H ) is called a subhypergroup of (H ) ifx K = K x = K for all x isin K If a subhypergroup is isomorphic to a group then we saythat it is a subgroup of (H )

Given a semihypergroup (H ) the relation βlowast in H is the transitive closure of therelation β =

⋃nge1 βn where β1 is the diagonal relation in H and for every integer n gt 1

βn is defined recursively as follows

xβny lArrrArr exist(z1 zn) isin Hn x y sube z1 z2 middot middot middot zn

We let βlowast(x) denote the βlowast-class of x isin H The relations β and βlowast are among thebest known fundamental relations [3] Their relevance in hyperstructure theory stemsfrom the following facts [2] If (H ) is a semihypergroup (respectively a hypergroup)then the quotient set Hβlowast equipped with the operation βlowast(x) otimes βlowast(y) = βlowast(z) for allx y isin H and z isin x y is a semigroup (respectively a group) Moreover the relation βlowast isthe smallest strongly regular equivalence on H such that the quotient Hβlowast is a semigroup(resp a group) The canonical projection ϕ H 7rarr Hβlowast is a good homomorphismthat is ϕ(x y) = ϕ(x) otimes ϕ(y) for all x y isin H The relations β and βlowast are also usefulto introduce notable families of semihypergroups and hypergroups including the fullysimple semihypergroups [18ndash20] and the 0-simple semihypergroups [1421ndash23] havinginteresting connections with partially ordered sets and integer sequences Furthermorewe recall from [810] that if (H ) is a hypergroup then β is transitive so that β = βlowast inevery hypergroup

If (H ) is a hypergroup then Hβlowast is a group and the kernel ωH = ϕminus1(1Hβlowast) ofϕ is the heart of (H ) Furthermore if |ωH | = 1 then (H ) is a 1-hypergroup For laterreference we collect in the following theorem a couple of classic results concerning theheart of a hypergroup see [24]

Theorem 1 Let (H ) be a hypergroup Then

1 β(x) = x ωH = ωH x for all x isin H2 (x y) capωH 6= emptylArrrArr (y x) capωH 6= empty for all x y isin H

140


If A is a non-empty set of a semihypergroup (H ) then we say that A is a completepart if it fulfills the following condition for every n isin Nminus 0 and (x1 x2 xn) isin Hn

(x1 middot middot middot xn) cap A 6= empty =rArr (x1 middot middot middot xn) sube A

For every non-empty set X of H the intersection of all the complete parts containingX is called the complete closure of X and is denoted with C(X) Clearly X is a completepart of (H ) if and only if C(X) = X If (H ) is a semihypergroup and ϕ H 7rarr Hβlowast isthe canonical projection then for all non-empty set A sube H we have C(A) = ϕminus1(ϕ(A))Moreover if (H ) is a hypergroup then

C(A) = ϕminus1(ϕ(A)) = A ωH = ωH A

A semihypergroup or hypergroup (H ) is complete if x y = C(x y) for all x y isin HIf (H ) is a complete (semi-)hypergroup then

x y = C(a) = βlowast(a)

for every x y isin H and a isin x y Recently Sonea and Cristea analyzed in [24] the commu-tativity degree of complete hypergroups stressing their similarities and differences withrespect to group theory The interested reader can find all relevant definitions propertiesand applications of hyperstructures and fundamental relations even in more abstractcontexts also in [425ndash30]

In what follows if (H ) is a finite hypergroup and |H| = n then we set H = 1 2 nMoreover if (H ) is a (possibly infinite) 1-hypergroup then we adopt the conventionωH = 1

3 Main Results

In this section we prove some results which will be used to classify the 1-hypergroupsof sizes 4 5 and 6 To this aim we now give a construction of hypergroups which undercertain conditions allows us to determine non-complete 1-hypergroups starting fromcomplete 1-hypergroups

31 A New Construction

Let (G middot) be a group with |G| ge 2 and let F = AkkisinG be a family of non-empty andpairwise disjoint sets indexed by G Let i j isin Gminus 1G be not necessarily distinct elementsand let ϕ Ai times Aj 7rarr Plowast(Aij) be any function such that for all a isin Ai and b isin Aj

⋃

xisinAj

ϕ(a x) =⋃

xisinAi

ϕ(x b) = Aij (1)

As a shorthand introduce the infix notation Ai times Aj 7rarr Aij defined by a b =ϕ(a b) for every a isin Ai and b isin Aj This operation is naturally extended to sets as followsfor X isin Plowast(Ai) and Y isin Plowast(Aj) let

a Y =⋃

yisinYa y X b =

⋃

xisinX= x b X Y =

⋃

xisinXyisinYx y

Hence the condition (1) can be reformulated as Ai b = a Aj = Aij Now letH =

⋃kisinG Ak and consider the hyperproduct H times H 7rarr Plowast(H) defined as follows for

all x y isin H let

x y =

Ars if x isin Ar y isin As and (r s) 6= (i j)x y if x isin Ai and y isin Aj

The following result shows the usefulness of this construction

141


Proposition 1 In the previous notation

1 for every r s isin G and x isin As we have Ar x = Ars and x Ar = Asr2 the hyperproduct is associative for every r s t isin G x isin Ar y isin As and z isin At we have

(x y) z = A(rs)t = Ar(st) = x (y z)

3 for every z1 z2 zn isin H with n ge 3 there exists r isin G such that z1 z2 middot middot middot zn = Ar4 (H ) is a hypergroup such that β = β25 for every x isin H x isin Ak lArrrArr β(x) = Ak6 Hβ sim= G and ωH = A1G 7 if |A1G | = 1 then (H ) is a 1-hypergroup8 (H ) is complete if and only if a b = Aij for every a isin Ai and b isin Aj

Proof In the stated hypothesis we have

1 Let r s isin G and x isin As If r 6= i or s 6= j then Ar x =⋃

yisinAr (y x) = Ars Otherwiseif r = i and s = j then Ar x = Ai x = Ai x = Aij by Equation (1) The identityx Ar = Asr can be derived by similar arguments

2 For every r s t isin G and x isin Ar y isin As and z isin At we have

(r s) 6= (i j) =rArr (x y) z = Ars z = A(rs)t

Moreover since j 6= 1G and the sets of the family F are pairwise disjoint if (r s) = (i j)then Aij 6= Ai and a z = A(ij)t = A(rs)t for every a isin x y sube Aij Therefore

(x y) z = (x y) z =⋃

aisinxya z = A(rs)t

The identity x (y z) = A(rs)t follows analogously3 It suffices to apply points 1 and 2 above and proceed by induction on n4 By 2 (H ) is a semihypergroup To prove that it is a hypergroup it remains to prove

that the hyperproduct is reproducible Let x isin H If x isin Ai then

x H =⋃

yisinHx y =

( ⋃

yisinAj

x y)⋃( ⋃

yisinHminusAj

x y)

=(

x Aj

)⋃( ⋃

risinGminusjAir

)= Aij cup (H minus Aij) = H

If x isin Ah with h 6= i then x H =⋃

yisinH x y =⋃

risinG Ahr = H because hG = GTherefore x H = H The identity H x = H can be shown analogously by consid-ering separately the cases x isin Aj and x isin H minus Aj Therefore is reproducible and(H ) is a hypergroup Consequently we have the chain of inclusions

β1 sube β2 sube middot middot middot sube βn sube middot middot middot

Now let x y isin H be such that xβy Hence there exists n ge 3 such that xβny Bypoint 3 there exists r isin G such that x y sube Ar For every a isin A1 we havex y sube Ar = x a and we obtain xβ2y

5 Let x isin Ak If a isin A1 then Ak = x a and so y isin Ak implies yβ2x Conversely ifyβ2x then there exist a b isin H such that x y sube a b From the definition of thehyperproduct it follows that there exists r isin G such that a b sube Ar Thereforesince x isin Ak cap Ar and the sets of the family F are pairwise disjoint we obtainy isin a b sube Ar = Ak Finally Ak = β(x) because β2 = β

6 The application f G 7rarr Hβ such that f (k) = Ak is a group isomorphism Moreoversince 1Hβ = f (1G) = A1G we conclude ωH = A1G

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7 The claim follows immediately from points 4 and 68 Trivial

We stress the fact that the hypothesis i j 6= 1G placed in the above construction isessential for the validity of Proposition 1 In fact if that hypothesis is not fulfilled thenthe hyperproduct defined by our construction may not be associative as shown by thefollowing example

Example 1 Let G sim= Z2 (i j) = (2 1) A1 = a b and A2 = c d Consider the functionϕ A2 times A1 7rarr Plowast(A2) represented by the following table

a b

c c dd d c

In this case the previous construction determines the following hyperproduct table

a b c d

a A1 A1 A2 A2b A1 A1 A2 A2c c d A1 A1d d c A1 A1

We have c A1 = d A1 = A2 and A2 a = A2 b = A2 hence the hyperproduct is notassociative because

(c a) a = c sub A2 c (a a) = c A1 = A2

Remark 1 The complete hypergroups have been characterized by Corsini in [4] by means of aconstruction very similar to ours In fact the above construction reduces to the one in [4] if thecondition in Equation (1) is replaced by ϕ(a b) = Aij for every a isin Ai and b isin Aj In that casethe hypergroup thus produced is complete

32 Auxiliary Results

Now we prove two results that are valid in every hypergroup Recall that in everyhypergroup the relation β is an equivalence coinciding with βlowast [810]

Proposition 2 Let (H ) be a hypergroup For all x y isin H x β(y) = β(x) β(y) = β(x) y

Proof By Theorem 1(1) we have x β(y) = x (ωH ωH y) = (x ωH) (ωH y) =β(x) β(y) = β(x) (ωH y) = (β(x) ωH) y = β(x) y

Proposition 3 Let (H ) be a hypergroup If a is an elements of H such that β(a) = a thenboth a b and b a are β-classes for all b isin H

Proof By Proposition 2 a b = β(a) b = β(a) β(b) The identity b a = β(b) β(a) isobtained analogously

The next results concern the properties of 1-hypergroups

Corollary 1 Let (H ) be a 1-hypergroup If there exists only one β-class of size greater than 1then H is complete

143


Proof Let β(x) be the only β-class with |β(x)| gt 1 By Proposition 3 we only have toprove that if a isin β(x) then both a b and b a are β-classes for all b isin H Let ϕ H 7rarr Hβbe the canonical projection and c isin a b We prove that a b = β(c) If |β(c)| = 1 thena b = β(c) If |β(c)| gt 1 then β(c) = β(x) = β(a) Consequently

ϕ(x) = ϕ(c) = ϕ(a)otimes ϕ(b) = ϕ(x)otimes ϕ(b)

hence ϕ(b) = 1Hβ and we have b isin ωH = 1 Finally a b = a 1 = β(a) = β(c)Analogous arguments can prove that also b a is a β-class

Remark 2 If H is not a complete 1-hypergroup and H owns exactly two β-classes β(a) and β(b)of size greater than 1 then β(a) β(a) = β(b) or β(b) β(b) = β(a)

From Corollary 1 we get the following results

Proposition 4 Let (H ) be a finite 1-hypergroup If |Hβ| = p and there exists a β-class ofsize |H| minus p + 1 then H is a complete hypergroup

The previous proposition allows us to find a simple proof to a result shown in [4]providing a taxonomy of all 1-hypergroups of size up to 4

Theorem 2 If (H ) is 1-hypergroup and |H| le 4 then (H ) is a complete hypergroup More-over (H ) is either a group or is one of the hypergroups described by the following three hyper-product tables up to isomorphisms

1 2 3

1 1 2 3 2 32 2 3 1 13 2 3 1 1

1 2 3 4

1 1 2 3 4 2 3 4 2 3 42 2 3 4 1 1 13 2 3 4 1 1 14 2 3 4 1 1 1

1 2 3 4

1 1 2 3 2 3 42 2 3 4 4 13 2 3 4 4 14 4 1 1 2 3

Proof Let (H ) be a 1-hypergroup of size le 4 that is not a group Two cases are possible(i) |H| = 3 and |Hβ| = 2 (ii) |H| = 4 and |Hβ| isin 2 3 In both cases (H ) is acomplete 1-hypergroup by Proposition 4 The corresponding hyperproduct tables arederived from Remark 1

Proposition 5 Let (H ) be a 1-hypergroup and let a b be elements of H such that β(a) β(b) = 1and β(a) β(a) = β(b) Then

1 β(b) β(a) = 1 and β(b) β(b) = β(a)2 if aprime aprimeprime isin β(a) and aprime aprimeprime = A then

(a) A x = x A = β(a) for all x isin β(b)(b) if there exist bprime bprimeprime isin β(b) such that bprime bprimeprime = aprime or bprime bprimeprime = aprimeprime then A = β(b)

144


Proof 1 The claim follows immediately from Theorem 12 (a) β(a) = β(aprime) = aprime 1 = aprime (aprimeprime x) = (aprime aprimeprime) x = A x and β(a) =

β(aprimeprime) = 1 aprimeprime = (x aprime) aprimeprime = x (aprime aprimeprime) = x A

(b) If bprime bprimeprime = aprime then A = aprime aprimeprime = (bprime bprimeprime) aprimeprime = bprime (bprimeprime aprimeprime) = bprime 1 = β(b)In the same way if bprime bprimeprime = aprimeprime then A = aprime aprimeprime = aprime (bprime bprimeprime) =(aprime bprime) bprimeprime = 1 bprimeprime = β(b)

In the forthcoming sections we will determine the hyperproduct tables of 1-hypergroupsof sizes 5 and 6 up to isomorphisms Since β is an equivalence the β-classes of a hyper-group (H ) determine a partition of H in disjoint subsets By Theorem 1(1) if (H ) is afinite 1-hypergroup such that H = 1 2 n and ωH = 1 then the first row and thefirst column of the hyperproduct table exhibits the sets of the partition In order to find the1-hypergroups of size n with |Hβ| = r we will consider all the non-increasing partitionsof the integer (nminus 1) in exactly (rminus 1) positive integers

4 1-Hypergroups of Size 5

In this section we determine the hyperproduct tables of 1-hypergroups of size 5 apartof isomorphisms Hence we put H = 1 2 3 4 5 and proceed with the analysis byconsidering the following cases corresponding to the non-increasing partitions of 4

1 |Hβ| = 2 β(2) = 2 3 4 52 |Hβ| = 3 β(2) = 2 3 4 β(5) = 53 |Hβ| = 3 β(2) = 2 3 β(4) = 4 54 |Hβ| = 4 β(2) = 2 3 β(4) = 4 β(5) = 55 |Hβ| = 5 and β(x) = x for all x isin H

Case 1 In the first case Hβ sim= Z2 so we only have the following complete hypergroup

1 1 2 3 4 5

1 1 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 52 2 3 4 5 1 1 1 13 2 3 4 5 1 1 1 14 2 3 4 5 1 1 1 15 2 3 4 5 1 1 1 1

Case 2 By Proposition 4(2) (H ) is a complete hypergroup and so its hyperproduct tableis the following apart of isomorphisms

2 1 2 3 4 5

1 1 2 3 4 2 3 4 2 3 4 52 2 3 4 5 5 5 13 2 3 4 5 5 5 14 2 3 4 5 5 5 15 5 1 1 1 2 3 4

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Case 3 Here Hβ sim= Z3 and setting β(2) = 2 3 and β(4) = 4 5 we derive thefollowing partial hyperproduct table

1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 1 13 2 3 1 14 4 5 1 15 4 5 1 1

By Proposition 5

bull if a b aprime bprime are elements in β(2) then

|a b| = |aprime bprime| = 1 =rArr 4 4 = 4 5 = 5 4 = 5 5 = 2 3



Therefore if we denote

P 2 3

23

Q 4 5

45

then we can restrict ourselves to the following three sub-cases

bull The tables P and Q do not contain any singleton entry Here one complete hyper-group arises

3 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 5 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

bull The table P contains (one or more) singleton entries in the main diagonal only Withoutloss of generality we can set 2 2 = 4 and obtain

P 2 3

2 4 4 53 4 5 R

Q 4 5

4 2 3 2 35 2 3 S

where R isin 4 5 4 5 and S isin 3 2 3 that is to say there are 6 tablesto examine Rejecting the hyperproduct tables that are not reproducible and theisomorphic copies we are left with the following 4 hypergroups

4 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 3

5 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

146


6 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

7 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 4 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

bull The table P contains at least one singleton entry off the main diagonal for instance2 3 = 4 Consequently from Proposition 5 we have

P 2 3

2 43

Q 4 5

4 2 3 2 35 2 3 2 3

where every empty cell can be filled with 4 or 5 or 4 5 giving rise to 27 moretables After checking reproducibility and isomorphisms we find the following8 hypergroups

8 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

9 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 5 4 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

10 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 5 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

11 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

12 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

13 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

14 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 5 4 1 13 2 3 4 5 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

15 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 5 4 1 13 2 3 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

Case 4 Here being |Hβ| = 4 three more 1-hypergroups are obtained by considering thatthe quotient group Hβ is isomorphic to either the group Z4 or the group Z2 timesZ2

147


bull If Hβ sim= Z4 and the β-class β(2) is associated with a generator of Z4 then we have

16 1 2 3 4 5

1 1 2 3 2 3 4 52 2 3 4 4 5 13 2 3 4 4 5 14 4 5 5 1 2 35 5 1 1 2 3 4

bull If Hβ sim= Z4 and the β-class β(2) is not associated with a generator of Z4 then we have

17 1 2 3 4 5

1 1 2 3 2 3 4 52 2 3 1 1 5 43 2 3 1 1 5 44 4 5 5 2 3 15 5 4 4 1 2 3

bull If Hβ sim= Z2 timesZ2 then we have

18 1 2 3 4 5

1 1 2 3 2 3 4 52 2 3 1 1 5 43 2 3 1 1 5 44 4 5 5 1 2 35 5 4 4 2 3 1

Case 5 Lastly in this case we have trivially H sim= Z5 as |Hβ| = 5

Therefore we have obtained the following result

Theorem 3 Apart of isomorphisms there are 19 1-hypergroups of size 5 Of these hypergroupsexactly 7 are complete

Remark 3 With the only exception of the hypergroup (H 4) in case 3 the 1-hypergroups ofsize 5 can be determined by the construction defined in Section 31 In fact the hypergroups(H k) with k isin 1 2 3 are also complete The hypergroups (H k) with k isin 5 6 middot middot middot 15are obtained by considering G sim= Z3 A1 = 1 A2 = 2 3 A3 = 4 5 and the functionsϕk A2 times A2 7rarr Plowast(A3) defined as ϕk(a b) = a k b for a b isin A2 and k isin 5 6 middot middot middot 15


In this section we classify the product tables of 1-hypergroups of size 6 apart ofisomorphisms Hence we assume H = 1 2 3 4 5 6 ωH = 1 and distinguish thefollowing nine cases

1 |Hβ| = 2 β(2) = 2 3 4 5 62 |Hβ| = 3 β(2) = 2 3 4 5 β(6) = 63 |Hβ| = 3 β(2) = 2 3 4 β(5) = 5 64 |Hβ| = 4 Hβ sim= Z4 β(2) = 2 3 4 β(5) = 5 β(6) = 65 |Hβ| = 4 Hβ sim= Z4 β(2) = 2 3 β(4) = 4 5 β(6) = 66 |Hβ| = 4 Hβ sim= Z2 timesZ2 β(2) = 2 3 4 β(5) = 5 β(6) = 67 |Hβ| = 4 Hβ sim= Z2 timesZ2 β(2) = 2 3 β(4) = 4 5 β(6) = 68 |Hβ| = 5 β(2) = 2 3 β(4) = 4 β(5) = 5 β(6) = 69 |Hβ| = 6

148


In all aforesaid cases except case 3 we can give the hyperproduct tables of the1-hypergroups apart of isomorphisms To achieve this goal we use the partition of Hinto β-classes the involved quotient group and the reproducibility condition that thehyperproduct tables must satisfy In case 3 we obtain a too high number of tables andit is impossible to list them Nevertheless with the help of a computer algebra systemwe are able to perform an exhaustive search of all possible hyperproduct tables and todetermine their number apart from isomorphisms To improve readability we postponethe discussion of case 3 at the end of this chapter

Case 1 The quotient group Hβ is isomorphic to Z2

1 2 3 4 5 6

1 1 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 62 2 3 4 5 6 1 1 1 1 13 2 3 4 5 6 1 1 1 1 14 2 3 4 5 6 1 1 1 1 15 2 3 4 5 6 1 1 1 1 16 2 3 4 5 6 1 1 1 1 1


1 2 3 4 5 6

1 1 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 62 2 3 4 5 6 6 6 6 13 2 3 4 5 6 6 6 6 14 2 3 4 5 6 6 6 6 15 2 3 4 5 6 6 6 6 16 6 1 1 1 1 2 3 4 5

Case 4 By Corollary 1 we obtain two complete non-isomorphic hypergroups In particularwhere the only β-class of size larger than 1 is associated to a generator of Z4 we have thefollowing hyperproduct table

1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 62 2 3 4 5 5 5 6 13 2 3 4 5 5 5 6 14 2 3 4 5 5 5 6 15 5 6 6 6 1 2 3 46 6 1 1 1 2 3 4 5

Instead if the only β-class of size larger than 1 is associated to a non-generator of Z4we obtain the following table

1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 62 2 3 4 1 1 1 6 53 2 3 4 1 1 1 6 54 2 3 4 1 1 1 6 55 5 6 6 6 2 3 4 16 6 5 5 5 1 2 3 4

149


Case 5 Considering that the group Z4 has only one element x of order 2 and that β(6) isthe only β-class of size 1 we have to examine two sub-cases depending on whether theclass β(6) is associated to the element x or not

1 |Hβ| = 4 Hβ sim= Z4 β(2) = 2 3 β(4) = 4 5 β(6) = 6 and β(6) associatedto the only element of Z4 having order two

2 |Hβ| = 4 Hβ sim= Z4 β(2) = 2 3 β(4) = 4 5 β(6) = 6 and β(6) associatedto a generator of Z4

In the first case we obtain a complete hypergroup

1 2 3 4 5 6

1 1 2 3 2 3 4 5 4 5 62 2 3 6 6 1 1 4 53 2 3 6 6 1 1 4 54 4 5 1 1 6 6 2 35 4 5 1 1 6 6 2 36 6 4 5 4 5 2 3 2 3 1

In the second case by using the multiplicative table of Z4 and the reproducibility ofH we obtain the following partial table

1 2 3 4 5 6

1 1 2 3 2 3 4 5 4 5 62 2 3 X Y 6 6 13 2 3 Z T 6 6 14 4 5 6 6 1 1 2 35 4 5 6 6 1 1 2 36 6 1 1 2 3 2 3 4 5

with X cup Y = Z cup T = X cup Z = Y cup T = 4 5 If we suppose that X isin 4 4 5 upto isomorphisms we obtain 12 hyperproduct tables corresponding to the following valuesof the sets X Y Z T

(1) X = 4 Y = 5 Z = 5 T = 4(2) X = 4 Y = 5 Z = 5 T = 4 5(3) X = 4 Y = 5 Z = 4 5 T = 4(4) X = 4 Y = 5 Z = 4 5 T = 4 5(5) X = 4 Y = 4 5 Z = 5 T = 4 5(6) X = 4 Y = 4 5 Z = 4 5 T = 4(7) X = 4 Y = 4 5 Z = 4 5 T = 5(8) X = 4 Y = 4 5 Z = 4 5 T = 4 5(9) X = 4 5 Y = 4 Z = 4 T = 4 5(10) X = 4 5 Y = 4 Z = 5 T = 4 5(11) X = 4 5 Y = 4 Z = 4 5 T = 4 5(12) X = 4 5 Y = 4 5 Z = 4 5 T = 4 5

Remark 4 The previous 12 hypergroups can be derived from the construction shown in Section 31where we let G sim= Z4 A1 = 1 A2 = 2 3 A3 = 4 5 A4 = 6 and ϕ A2 timesA2 7rarr Plowast(A3) is the function defined as ϕ(a b) = a k b for a b isin A2 and k isin 1 2 12Incidentally we note that the hypergroup arising from 12 is also complete

150


Case 6 In this case we obtain only one 1-hypergroup which is also complete

1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 62 2 3 4 1 1 1 6 53 2 3 4 1 1 1 6 54 2 3 4 1 1 1 6 55 5 6 6 6 1 2 3 46 6 5 5 5 2 3 4 1

Case 7 In this case we also obtain only one 1-hypergroup which is also complete

1 2 3 4 5 6

1 1 2 3 2 3 4 5 4 5 62 2 3 1 1 6 6 4 53 2 3 1 1 6 6 4 54 4 5 6 6 1 1 2 35 4 5 6 6 1 1 2 36 6 4 5 4 5 2 3 2 3 1

Case 8 Here the quotient group is isomorphic to Z5 and we deduce one complete hypergroup

1 2 3 4 5 6

1 1 2 3 2 3 4 5 62 2 3 4 4 5 6 13 2 3 4 4 5 6 14 4 5 5 6 1 2 35 5 6 6 1 2 3 46 6 1 1 2 3 4 5

Case 9 Here β(x) = x forallx isin 2 3 4 5 6 and so H is a group of order 6 that is H sim= Z6or H sim= S3

To conclude the review of 1-hypergroups of size 6 hereafter we consider the mostchallenging case where a very high number of tables arises

Case 3 Here the quotient group Hβ is isomorphic toZ3 β(2) = 2 3 4 and β(5) = 5 6In this case there is only one complete 1-hypergroup its multiplicative table is the following

1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 6 5 62 2 3 4 5 6 5 6 5 6 1 13 2 3 4 5 6 5 6 5 6 1 14 2 3 4 5 6 5 6 5 6 1 15 5 6 1 1 1 2 3 4 2 3 46 5 6 1 1 1 2 3 4 2 3 4

In order to find the other 1-hypergroups we make sure that the sub-cases we aredealing with are disjoint from each other which means that a hypergroup of a sub-case cannot be isomorphic to a hypergroup of another sub-case

151


If (H ) is not a complete hypergroup then we can start from the partial table

1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 6 5 62 2 3 4 1 13 2 3 4 1 14 2 3 4 1 15 5 6 1 1 16 5 6 1 1 1

and the partial sub-tables

P

2 3 4

234

Q 5 6

56

Taking into account Proposition 5 there are three options

1 In the partial table Q there is at least one hyperproduct which is a singleton forinstance 2 and for all a aprime isin 2 3 4 we have a aprime = 5 6 We consider twosub-cases

(1a) the singleton can appear only in the main diagonal

Q 5 6

5 2 R6 S T

By reproducibility we have R S isin 3 4 2 3 4 and T isin Plowast(2 3 4)This yields 22 middot 7 = 28 tables to examine

(1b) The singleton must appear off the main diagonal

Q 5 6

5 R 26 T S

with R S isin 3 4 2 3 4 and T isin Plowast(2 3 4) Thus other 22 middot 7 = 28tables arise

2 The partial table Q contains at least one hyperproduct of size two for instance 2 3but there are no singletons inside Q Moreover for all a aprime isin 2 3 4 we havea aprime = 5 6 We obtain two subcases again

(2a) the hyperproduct 2 3 can appear only in the main diagonal

Q 5 6

5 2 3 2 3 46 2 3 4

and 6 6 isin 2 3 2 4 3 4 2 3 4 Hence 4 cases tables arise

152


(2b) the hyperproduct 2 3must appear out of the main diagonal

Q 5 6

5 R 2 36 S T

the hyperproducts R and T belong to the set 2 4 3 4 2 3 4 andS isin 2 3 2 4 3 4 2 3 4 Therefore 32 middot 4 = 36 cases arise

3 The partial table P contains at least one singleton Without loss in generality we cansuppose that 5 is among them From Proposition 5 we deduce 5 5 = 5 6 =6 5 = 2 3 4 The following two possibilities arise

(3a) Singletons can appear only in the main diagonal of P Therefore we put2 2 = 5 and obtain

P

2 3 4

2 5 5 6 5 63 5 6 R 5 64 5 6 5 6 S

Q 5 6

5 2 3 4 2 3 46 2 3 4 T

where R S isin 5 6 5 6 Moreover from Proposition 5 we deduce thatT 6= 2 that is T isin Plowast(2 3 4)minus 2 and 32 middot 6 = 54 cases arise

(3b) There is a singleton cell off the main diagonal of P for instance 2 3 = 5We obtain

P

2 3 4

2 534

Q 5 6

5 2 3 4 2 3 46 2 3 4 R

We consider two sub-cases

i R = 2 3 4 the 8 empty cells in table P can be filled with either 5or 6 or 5 6 Hence 38 cases arise

ii |R| lt 3 from Proposition 5 R 6= 2 R 6= 3 and so R isin 4 2 32 4 3 4 Moreover the table P can not contain the hyperproduct6 that is every cell in P has to be filled with 5 or 5 6 Thus28 middot 4 = 512 cases arise

All the previous sub-cases have been examined with the help of a computer algebrasystem based on MATLAB R2018a running on an iMac 2009 with an Intel Core 2 processor(306 GHz 4 GB RAM) The complete enumeration of all 1-hypergroups in case 3 tookabout 2 min utilizing the subdivision into sub-cases described above while without thatsubdivision the running time for solving case 3 exceeded 90 min We report in Table 1 thenumber of 1-hypergroups found in each sub-case considered above up to isomorphisms

Table 1 Number of non-isomorphic non-complete 1-hypergroups found in case 3 |H| = 6

Case (1a) (1b) (2a) (2b) (3a) (3b) Total

Hypergroups 13 13 3 12 12 1180 1233

Remark 5 The 1-hypergroups in sub-cases (1a) (1b) (2a) and (2b) can be derived from theconstruction shown in Section 31 where G sim= Z3 A1 = 1 A2 = 5 6 A3 = 2 3 4 andϕ A2 times A2 7rarr Plowast(A3) is the function defined by the corresponding partial tables Q

153


In Table 2 we summarize the results obtained in our case-by-case review of 1-hypergroupsof order 6 In that table we report the number of 1-hypergroups found in each case and thenumber of complete hypergroups among them Theorem 4 states the conclusion

Table 2 Number of non-isomorphic 1-hypergroups |H| = 6

Case 1 2 3 4 5 6 7 8 9 Total

Hypergroups 1 1 1234 2 13 1 1 1 2 1256Complete 1 1 1 2 2 1 1 1 2 11

Theorem 4 Up to isomorphisms there are 1256 1-hypergroups of size 6 of which 11 are complete

6 Conclusions and Directions for Further Research

A 1-hypergroup is a hypergroup (H ) where the kernel of the canonical projectionϕ H 7rarr Hβ is a singleton In this paper we enumerate the 1-hypergroups of size 5 and6 The main results are given in Theorem 3 for |H| = 5 and Theorem 4 for |H| = 6 Inparticular in Section 4 we show a representation of the 19 1-hypergroups of size 5 Toachieve this goal we exploit the partition of H induced by β In this way we reduce theanalysis of a tough problem to that of a few sub-problems that can be solved explicitly orby means of scientific computing software on an ordinary desktop computer Moreoverin Section 31 we give a construction of hypergroups which under certain conditions are1-hypergroups That construction is very flexible and many 1-hypergroups of size 5 and 6can be determined in that way

To highlight a direction for possible further research we point out that many hyper-groups found in the present work are also join spaces or transposition hypergroups To beprecise let (H ) be a hypergroup and for every a b isin H let ab and ba denote the setsx isin H | a isin x b and x isin H | a isin b x respectively The commutative hypergroupsfulfilling the transposition axiom that is

ab cap cd 6= empty =rArr a d cap b c 6= empty

for all a b c d isin H are called join spaces These hypergroups have been widely usedin Geometry [3132] In [33] Jantosciak generalized the transposition axiom to arbitraryhypergroups as follows

ba cap cd 6= empty =rArr a d cap b c 6= empty

for all a b c d isin H These particular hypergroups are called transposition hypergroupsA number of results on transposition hypergroups can be found in eg [33ndash35] Forexample it is known that the complete hypergroups are also transposition hypergroupsThe construction shown in Section 31 produces transposition hypergroups when a dcap b c 6= empty for all a b isin Ai and c d isin Aj Indeed if x isin ba cap cd then a isin b x and c isin x dThus we have a d cup b c sube b x d By point 3 of Proposition 1 there exists k isin Gsuch that b x d = Ak By definition of if k 6= ij then a d = b c = Ak Otherwise ifk = ij then we have a b isin Ai c d isin Aj a d = a d and b c = b c Hence by hypotesisa d cap b c 6= empty

Based on the preceding comment we plan to characterize and enumerate the1-hypergroups of small size that also are join spaces or transposition hypergroups infurther works

Author Contributions Conceptualization investigation writingmdashoriginal draft MDS DF(Domenico Freni) and GLF software writingmdashreview and editing DF (Dario Fasino) Allauthors have read and agreed to the published version of the manuscript

154


Funding The research work of Mario De Salvo was funded by Universitagrave di Messina Italy grantFFABR Unime 2019 Giovanni Lo Faro was supported by INdAM-GNSAGA Italy and by Universitagravedi Messina Italy grant FFABR Unime 2020 The work of Dario Fasino was partially supported byINdAM-GNCS Italy


References1 Massouros G Massouros C Hypercompositional algebra computer science and geometry Mathematics 2020 8 1338 [CrossRef]2 Koskas H Groupoiumldes demi-hypergroupes et hypergroupes J Math Pures Appl 1970 49 155ndash1923 Vougiouklis T Fundamental relations in hyperstructures Bull Greek Math Soc 1999 42 113ndash1184 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Tricesimo Italy 19935 Davvaz B Salasi A A realization of hyperrings Commun Algebra 2006 34 4389ndash4400 [CrossRef]6 De Salvo M Lo Faro G On the nlowast-complete hypergroups Discret Math 1999 208209 177ndash188 [CrossRef]7 De Salvo M Fasino D Freni D Lo Faro G Semihypergroups obtained by merging of 0-semigroups with groups Filomat 2018

32 4177ndash41948 Freni D Une note sur le cœur drsquoun hypergroup et sur la clocircture transitive βlowast de β Riv Mat Pura Appl 1991 8 153ndash1569 Freni D Strongly transitive geometric spaces Applications to hypergroups and semigroups theory Commun Algebra 2004 32

969ndash988 [CrossRef]10 Gutan M On the transitivity of the relation β in semihypergroups Rend Circ Mat Palermo 1996 45 189ndash200 [CrossRef]11 Norouzi M Cristea I Fundamental relation on m-idempotent hyperrings Open Math 2017 15 1558ndash1567 [CrossRef]12 De Salvo M Fasino D Freni D Lo Faro G On hypergroups with a β-class of finite height Symmetry 2020 12 168 [CrossRef]13 Sadrabadi E Davvaz B Atanassovrsquos intuitionistic fuzzy grade of a class of non-complete 1-hypergroups J Intell Fuzzy Syst

2014 26 2427ndash2436 [CrossRef]14 Fasino D Freni D Fundamental relations in simple and 0-simple semi-hypergroups of small size Arab J Math 2012 1 175ndash190

[CrossRef]15 Freni D Minimal order semi-hypergroups of type U on the right II J Algebra 2011 340 77ndash89 [CrossRef]16 Heidari D Freni D On further properties of minimal size in hypergroups of type U on the right Comm Algebra 2020 48

4132ndash4141 [CrossRef]17 Fasino D Freni D Existence of proper semi-hypergroups of type U on the right Discrete Math 2007 307 2826ndash2836 [CrossRef]18 De Salvo M Fasino D Freni D Lo Faro G Fully simple semihypergroups transitive digraphs and Sequence A000712

J Algebra 2014 415 65ndash87 [CrossRef]19 De Salvo M Freni D Lo Faro G Fully simple semihypergroups J Algebra 2014 399 358ndash377 [CrossRef]20 De Salvo M Freni D Lo Faro G Hypercyclic subhypergroups of finite fully simple semihypergroups J Mult Valued Log Soft

Comput 2017 29 595ndash61721 De Salvo M Fasino D Freni D Lo Faro G A family of 0-simple semihypergroups related to sequence A00070 J Mult Valued

Log Soft Comput 2016 27 553ndash57222 De Salvo M Freni D Lo Faro G On hypercyclic fully zero-simple semihypergroups Turk J Math 2019 43 1905ndash1918

[CrossRef]23 De Salvo M Freni D Lo Faro G On further properties of fully zero-simple semihypergroups Mediterr J Math 2019 16 48

[CrossRef]24 Sonea A Cristea I The class equation and the commutativity degree for complete hypergroups Mathematics 2020 8 2253

[CrossRef]25 Antampoufis N Spartalis S Vougiouklis T Fundamental relations in special extensions In Algebraic Hyperstructures and

Applications Vougiouklis T Ed Spanidis Press Xanthi Greece 2003 pp 81ndash8926 Davvaz B Semihypergroup Theory Academic Press London UK 201627 Davvaz B Leoreanu-Fotea V Hyperring Theory and Applications International Academic Press Palm Harbor FL USA 200728 De Salvo M Lo Faro G A new class of hypergroupoids associated with binary relations J Mult Valued Log Soft Comput 2003

9 361ndash37529 Massouros GG The subhypergroups of the fortified join hypergroup Ital J Pure Appl Math 1997 2 51ndash6330 Naz S Shabir M On soft semihypergroups J Intell Fuzzy Syst 2014 26 2203ndash2213 [CrossRef]31 Prenowitz W A contemporary approach to classical geometry Am Math Month 1961 68 1ndash67 [CrossRef]32 Prenowitz W Jantosciak J Join Geometries A Theory of Convex Sets and Linear Geometry Springer New York NY USA 197933 Jantosciak J Transposition hypergroups Noncommutative join spaces J Algebra 1997 187 97ndash119 [CrossRef]34 Jantosciak J Massouros CG Strong identities and fortification in transposition hypergroups J Discrete Math Sci Cryptogr

2003 6 169ndash193 [CrossRef]35 Massouros CG Massouros GG The transposition axiom in hypercompositional structures Ratio Math 2011 21 75ndash90

155

mathematics

Article

n-Ary Cartesian Composition of Multiautomatawith Internal Link for Autonomous Control ofLane Shifting

Štepaacuten Krehliacutek

CDVmdashTransport Research Centre Liacutešenskaacute 33a 636 00 Brno Czech Republic stepankrehlikcdvczTel +420-724-863-410

Received 27 April 2020 Accepted 19 May 2020 Published 21 May 2020

Abstract In this paper which is based on a real-life motivation we present an algebraic theory ofautomata and multi-automata We combine these (multi-)automata using the products introducedby W Doumlrfler where we work with the cartesian composition and we define the internal linksamong multiautomata by means of the internal linksrsquo matrix We used the obtained product of n-arymulti-automata as a system that models and controls certain traffic situations (lane shifting) forautonomous vehicles

Keywords hypergroup automata theory cartesian composition autonomous vehicles cooperativeintelligent transport systems

1 Introduction

11 Motivation

Every physical object in real time and space is defined by its specific properties such as itsposition in spacetime temperature shape or dimension This set of properties may be regarded asa state in which the physical object evinces While focusing on transport infrastructure there are a lotof elements of characteristic properties These characteristics are considered as states of elementsin the transport infrastructure for example a state of traffic lights a state of velocity a state ofmileage or a state of traffic density A clear description of such states is not the only requirementThe possibility of a state change is an equally important requirement In this respect the algebraictheory of automata suggests a specific tool (automata or multi-automata) that enables to changea state using the transition function and input words In other words the input symbol (or inputword) is applied to a state by a transition function and consequently a new state is obtained In thepast these structures were considered as systems that transmitted a specific type of informationNowadays this concept cannot sufficiently describe or even control real-life applications as theseare too complex In such complex and difficult systems a change of a state causes a change ofanother state for example a train that passes a railroad crossing stops vehicles on a road In currenttraffic a driver of a vehicle usually responds to the change in the state of the surrounding elementsin transportmdashgenerally this is what we consider a human factor Traffic situations will get morecomplicated when autonomous vehicles or even whole autonomous systems that control the trafficare included In the case of autonomous systems we suggest that every vehicle communicates withother vehicles or the infrastructure and as a result it influences the next state of a system The fact thatautonomous vehicles can communicate and send information about their state or information abouta planned change of their state is considered as an advantage that makes the traffic infrastructure moreeffective In this paper we construct a cartesian composition of multi-automata ie we combine somemulti-automata (or automata) and we add internal links between their particular states This new

157


approach to a system of multi-automata with an internal link (SMAil) is defined by matrices andenables to describe and control the above-mentioned problems

12 Basic Terminology of the Hyperstructure Theory

Before we introduce a theory of automata recall some basic notions of the algebraichyperstructure theory (or theory of algebraic hypercompositional structures) For further reference seefor example books [12] A hypergroupoid is a pair (H lowast) where H is a nonempty set and the mappinglowast HtimesH minusrarr Plowast(H) is a binary hyperoperation (or hypercomposition) on H (here Plowast(H) denotes thesystem of all nonempty subsets of H) If a lowast (b lowast c) = (a lowast b) lowast c holds for all a b c isin H then (H lowast) iscalled a semihypergroup If moreover the reproduction axiom ie relation a lowast H = H = H lowast a for alla isin H is satisfied then the semihypergroup (H lowast) is called hypergroup By extensive hypergroup inthe sense of hypercompositional structures we mean a hypergroup that a b sube a lowast b for all a b isin Hie that the elements are included in its ldquoresalt of hyperoperationrdquo For the same application on thistheory in other science or real-life problems see for example [3ndash7]

13 Some Notions of the (Multi-)Automata Theory

In the field of automata theory several types of automata are considered In this work we focuson automata without outputs ie a structure composed from a triad of an input alphabet a set of statesand a transition function For completeness let us note that the automaton is a quintuple consisting ofthe above-mentioned triad plus an output alphabet and an output function For more details see [8ndash11]For details regarding terminology and some minor deviations from standard usage see Novaacutek Krehliacutekand Stanek [12] Further on we recall the following definition (notice that condition 1 is sometimesomitted if we regard a semigroup instead of a monoid)

Definition 1 ([13]) By a quasindashautomaton we mean a structure A = (I S δ) such that I 6= empty is a monoidS 6= empty and δ I times Srarr S satisfies the following condition

1 There exists an element e isin I such that δ(e s) = s for any state s isin S2 δ(y δ(x s)) = δ(xy s) for any pair x y isin I and any state s isin S

The set I is called the input set or input alphabet the set S is called the state set and the mapping δ is callednext-state or transition function Condition 1 is called the unit condition (UC) while condition 2 is called theMixed Associativity Condition (MAC)

Next we are going to work with the idea of a quasi-multiautomaton as a hyperstructuregeneralization of a quasi-automaton see [14ndash18] When adjusting the conditions imposed on thetransition function δ it must be defined cautiously because we get a state on the left-hand side ofDefinition 1 condition 2 whereas we get a set of states on the right-hand side

Definition 2 ([13]) A quasindashmultiautomaton is a triad MA = (I S δ) where (I lowast) is a semihypergroup S isa non-empty set and δ I times Srarr S is a transition function satisfying the condition

δ(b δ(a s)) isin δ(a lowast b s) for all a b isin I s isin S (1)

The semihypergroup (I lowast) is called the input semihypergroup of the quasindashmultiautomaton A (I alone iscalled the input set or input alphabet) the set S is called the state set of the quasindashmultiautomaton A and δ iscalled next-state or transition function Elements of the set S are called states elements of the set I are calledinput symbols or letters Condition (1) is called Generalized Mixed Associativity Condition (GMAC)

In the theory of automata ways to combine automata into one structure have been describedWe will recall homogeneous heterogeneous products and cartesian composition which was introduced by [9]These products were constructed and investigated primarily on classical automata without output

158


in [19] In the following definition we recall all three types of productscompositions introduced byW Doumlrfler

Definition 3 ([20]) Let A1 = (I S δ) A2 = (I R τ) and B = (J T σ) be quasindashautomata By thehomogeneous product A1 timesA2 we mean the quasindashautomaton (I Stimes R δtimes τ) where δtimes τ I times (Stimes R)rarrStimes R is a mapping satisfying for all s isin S r isin R a isin H (δtimes τ)(a (s r)) = (δ1(a s) τ(a r)) while theheterogeneous product A1otimesB is the quasi-automaton (Itimes J Stimes T δotimes σ) where δotimes τ (Itimes J)times (Stimes T)rarrStimesT is a mapping satisfying for all a isin I b isin J s isin S t isin T δotimes σ((a b) (s t)) = (δ(a s) σ(b t)) For I Jdisjoint we by A middotB denote the cartesian composition of A and B ie the quasindashautomaton (I cup J Stimes T δ middot σ)where δ middot σ (I cup J)times (Stimes T)rarr Stimes T is defined for all x isin I cup J s isin S and t isin T by

(δ middot σ) (x (s t)) =

(δ(x s) t) if x isin I

(s σ(x t)) if x isin J

One can see that in the homogeneous product we have the same input set which operates on everycomponent of the state set In the heterogeneous product the input is a pair of symbols such that eachinput symbol from the pair is operated on the respective component of the state pair In the cartesiancomposition the input is directed to the corresponding state and operates on one component of thestate pair only

Notice that generalizing the homogeneous or heterogeneous product of automata to the caseof quasi-multiautomata is rather straightforward see Chvalina Novaacutek and Krehliacutek [20] Howeverapplying the GMAC condition which distinguishes the transition function of a quasi-multiautomatonfrom a transition function of a quasi-automaton on cartesian composition of quasi-multiautomata isnot straightforward In Chvalina Novaacutek and Krehliacutek [13] two extensions of this condition

bull Extension Generalized Mixed Associativity Condition (E-GMAC)bull Small Extension Generalized Mixed Associativity Condition (SE-GMAC)

and their applications were suggested

Definition 4 ([13]) Let A = (I S δ)B = (J T σ) be e-quasi-multiautomata with input semihypergroups(I) (J lowast) and transition maps δ I times Srarr S σ J times T rarr T satisfying conditions

δ(y δ(x s)) isin δ(x middot y s) cup δ(I s) for all x y isin I s isin S (2)

σ(y σ(x t)) isin σ(x middot y t) cup σ(J t) for all x y isin J t isin T (3)

By the cartesian composition of e-quasi-multiautomata A and B denoted as A middotE B we mean thee-quasi-multiautomaton A middotE B = (((I cup J) ) S times T δ middot σ) where δ middot σ is for all x isin I cup J s isin S andt isin T defined by

(δ middot σ)(x (s t)) =


(s σ(x t)) if x isin J(4)

and (I cup J)times (I cup J)rarr Plowast(I cup J) is for all x y isin I cup J defined by

x y =

x y sube I if x y isin I

x lowast y sube J if x y isin J

x y if x isin I y isin J or x isin J y isin I

and δ middot σ (I cup J)times (Stimes T)rarr (Stimes T)

159


When modelling the process of data aggregation from underwater wireless sensor networksKrehliacutek and Novaacutek [21] suggested that using an internal link between respective automata in theircartesian composition describes the real-life context better because the internal link influences notonly the given individual automaton but to a certain extent the whole system In this paper we makeuse of [21] Therefore we include the following definition For an example explaining the concept(together with calculations) see [21] Example 1

Definition 5 ([21]) Let A = (I S δA) and B = (J T δB) be two quasindashautomata δA I times S rarr S δB J times T rarr T with disjoint input sets I J By A middot B we denote the automaton A middot B = (I cup J Stimes T δA middot δB)where δA middot δB (I cup J)times (Stimes T)rarr Stimes T and ϕ Srarr J $ T rarr I is defined by

(δA middot δB) (x (s t)) =

(δA(x s) δB(ϕ(δA(x s) t))) if x isin I

(δA($(δB(x t) s)) δB(x t)) if x isin J(5)

for all x isin I cup J s isin S and t isin T The quasindashautomaton A middot B is called the cartesian composition ofquasi-automata A and B with an internal link If in Definition 4 we replace Condition (4) with (5) we call theresulting quasi-multiautomaton the cartesian composition of quasindashmultiautomata A and B with an internal link

2 New Theoretical Model

In the classical theory automata were considered as systems for transfering information of specifictypes However given complicated systems used nowadays their benefits may not seem sufficientIn [9131920] we presented some real-life applications In this section we introduce an extension ofideas first included in [21] regarding internal links in the cartesian composition of quasi-multiautomataWe discuss systems in which there is a whole set of internal links For our theoretical purposes we willorganize them in a matrix considered by for example Golestan et al [22] In Section 3 we show theapplication of our theoretical results in the context of autonomous cars and their navigation

Notation 1 We are going to use the following notation

Cartesian product o f state sets notimes

i=1Si = S1 times S2 times S3 times times Sn

Cartesian product o f transiton f unctions nprodi=1

δi = δ1 middot δ2 middot δ3 middot δn

Definition 6 Let MAi = (Ii Si δi) be e-quasi-multiautomata with input semihypergroups (Ii lowasti) andδi Ii times Si rarr Si satisfying condition

δi(y (δi(x s)) isin δi(y lowasti x s) cup δi(Ii s) for all x y isin Ii s isin Si i isin 1 2 n (6)

By n-ary cartesian composition of the e-quasi-multiautomata we mean the following system ofe-quasi-multiautomata with internal link (SMAil)

Ancc =

((n⋃

i=1

Ii )

notimes

i=1

Sin

prodi=1

δi Mnn(ϕ)

)

wherenprodi=1

δi n⋃

i=1Ii times

notimesi=1

Si rarrnotimes

i=1Si and matrix Mnn(ϕ) called matrix of internal links where ϕij Si rarr Ij

for i j isin 1 2 n ie

160


Mnn(ϕ) =

ϕ11 ϕ12 ϕ1nϕ21 ϕ22 ϕ2n

ϕn1 ϕn2 ϕnn

is defined by

(n

prodi=1

δi

)(x (s1 s2 sn)) = (δ1 middot δ2 middot middot δn) (x (s1 s2 sn)) =

=

(δ1(ϕ11(δ1(x s1)) s1) δ2(ϕ12(δ1(x s1)) s2) δn(ϕ1n(δ1(x s1) sn)) if x isin I1

(δ1(ϕ21(δ2(x s2)) s1) δ2(ϕ22(δ2(x s2)) s2) δn(ϕ2n(δ2(x s2)) sn)) if x isin I2

(δ1(ϕn1(δn(x sn)) s1) δ2(ϕn2(δn(x sn)) s2) δn(ϕnn(δn(x sn)) sn) if x isin In

(7)

and n⋃

i=1Ii times

n⋃i=1

Ii rarr Plowast(

n⋃i=1

Ii

)is for all x y isin

n⋃i=1

Ii defined by

x y =

x lowasti y sube Ii if x y isin Ii

x y if x isin Ii y isin Ik where i 6= k

satisfies the condition(

n

prodi=1

δi

)(y

(n

prodi=1

δi

)(x (s1 s2 sn))

)isin(

n

prodi=1

δi

)(x y (s1 s2 sn)) cup

notimes

i=1

δi(Ii si) (8)

One can easily explain what is meant by Definition 6 using Figure 1 When an arbitrary input

x isinn⋃

i=1Ii is applied the system ie the cartesian composition has to find out in which input set x

belongs Therefore we determine the correct line in Equation (7) which through using we obtaina new state of the system If for example x isin I1 the new state sprime1 isin S1 will be computed Then it willbe adjusted to sprimeprime1 isin S1 using ϕ11 ie by the self-mapping internal link This new state sprimeprime1 isin S1 will bemapped to remaining input sets I2 I3 as defined by the matrix of internal links In other wordsthe matrix determines which other states can be influenced by the change of the primary state Weapply inputs on states using respective transition functions If there is 0 instead of ϕij in the matrixof internal links then the link between the state set Sij and input set Ij does not exist Notice thatin Equation (7) eg δ1(ϕn1(δn(x sn)) s1) = δ1(ϕn1(sprime) s1) = δ1(i1 s1) = sprimeprime Since we obviously haveto regard directions (see the oriented arrows) the matrix of internal links is not symmetric

In Section 3 we apply the matrix of internal links in the context of modelling the navigationof autonomous vehicles In this context each vehicle obviously adjusts its behaviour based on thebehaviour of other vehicles Also obviously not all vehicles need to be autonomous which explainsthe use of zeros in the matrix (only autonomous vehicles communicate)

The above reasoning can be seen applied in the following example Notice that we will use it alsofor the construction of the state set in Theorems 2 and 4 of Section 3

Example 1 Consider a section of a road in front of an autonomously controlled intersection in Figure 2All vehicles are autonomous with compatible self-driving control parameters Every car is in a certain state sfor example car A3 is in state s = (3 45 250 1 0 0) which means for example that car number 3 going at45 kmh is in the position 250 m from the intersection in the 1st lane from the right The first 0 means no changein speed while the second 0 means that the car is about to turn left

In our system it is obvious that vehicle A3 does not need to communicate with vehicle A4 because it isnot an obstacle in its intended driving operation Thus introducing an internal link between A3 and A4 is

161


not necessary However vehicles A1 and A2 are problematics for A3 because it intends to turn left ThereforeA1 should increase its speed while A2 should slow down

S

S

S

S

3

2

1

n

Ii

i=1

I

I

I

I

1

2

3

n

Input forSMAIL

Figure 1 SMAilmdashA system of multiautomata with internal links

A

A

AA

1

23

4

Figure 2 Lane shifting

Remark 1 Notice that some theoretical requests in the above example are not possible or even absurd iewe have to keep in mind their feasibility We already faced this problem in [1223] where it was eliminated byusing a suitable state set input set or operations on them In this paper we will choose special input and statesets as well

3 SMAilmdashApplication of the Theoretical Model for Autonomous Cooperation

We are going to construct our system called SMAil for the above context of Example 1 NaturallyExample 1 is an example of a possible usage of SMAil only

First we need to define suitable sets SpRM and Ip

VM where p isin N is the index The elements ofthe state set are ordered pairsmdashthe first component being an ordered sextuple of numbers while thesecond component is a matrix The state set is

SpRM =

[s A] |s isin Rp

6 A isin Mmn

(9)

where

Rp6 =

s = (s1 s2 s6)|s1 = p s2 s3 s5 isin R+

0 s4isin N s6 isin 〈minusπ

4

π

4〉

(10)

162


and

Mmn =

a11 a1n

am1 amn

|aij isin

0 for i = m+1

2 j = n+12

minus1 0 1 for otherwise

(11)

where m n are a odd numbers The state set describes the complete position of every vehicle inall lanes Notice that this is the same approach as in [2425] Now the elements of Rp

6 defined byEquation (10) correspond to states mentioned in Example 1 ie s1 stands for the vehicle number s2

for its speed s3 for distance from the intersection s4 for the respective lane (calculated from the rightusing odd numbers only even number are reserved for positions between two lanes) s5 stands forchanging velocity (ie an interval (0 1) is deceleration an interval (1 infin) is acceleration 1 stands forconstant speed) s6 stands for changing direction (ie an interval [minusπ

4 0) means manoeuvring leftinterval (0 π

4 ] means manoeuvring right 0 means straight direction) Of course more parameterscan be used for an example see [26] Matrices A used in Equation (9) are taken from the set Mmn

defined by Equation (11) Notice that we use these matrices in the form suggested in [252728] wherefor the purpose of its control policy the intersection is divided into a grid of reservation tiles To bemore precise

A =

minus1 minus1 minus1 minus1 minus1 minus1 1 1 11 1 1 minus1 minus1 minus1 1 1 11 1 1 minus1 minus1 minus1 minus1 minus1 minus11 1 1 minus1 minus1 minus1 minus1 minus1 minus1minus1 minus1 minus1 0 0 0 minus1 minus1 minus1minus1 minus1 minus1 0 0 0 1 1 1minus1 minus1 minus1 0 0 0 1 1 1minus1 minus1 minus1 minus1 minus1 minus1 1 1 1minus1 minus1 minus1 minus1 minus1 minus1 1 1 1minus1 minus1 minus1 minus1 minus1 minus1 1 1 1minus1 minus1 minus1 minus1 minus1 minus1 minus1 minus1 minus1

isin Mmn (12)

where 0 are parts of the grid (or tiles) occupied by the given vehicle 1 stands for tiles occupied by someother vehicles and minus1 are free tiles The above matrix A describes the situation depicted in Figure 3from the point of view of A3

Next for construction of the i-th quasi-multiautomaton and consequently SMAil we will needan input alphabet ie input sets (since we will be working with quasi-multiautomata these will bealgebraic hyperstructures) As mentioned above we will define Ip

VM where p isin N is the index of theinput set

IpVM =

[~i Cmn

]=

(i1 i2 i6)

c11 c1n

cm1 cmn

|i1 = p i2 isin R+0 i3 isin 〈0 1〉 i4 isin N0 i5 isin R+ i5 isin

langminusπ

4

π

4

rang

cij isin minus1 1 i isin 1 2 n j isin 1 2 m

(13)

163


AA

AA

1

2

3

4

123456789

1011

1 2 3 4 5 6 7 8 9Figure 3 Lane shifting grid view of vehicle A3

It is obvious that input sets IpVM and Iq

VM are disjunct for p 6= q because the first componentof every vector used in [~v Anm] isin Ip

VM is p while the first component of every vector used in[~u Bnm] isin Iq

VM is q ie

IpVM cap Iq

VM = empty for every p q isin N where p 6= q

Since vector components are real (or even natural) numbers we can suppose that their sets areordered we write le for this ordering For all p isin N we define hyperoperation lowastp Ip

VM times IpVM rarr

Plowast(IpVM) by

[~u Amn] lowastp [~v Bmn] = (14)

[~w Cmn] | minui vi le wi le maxui vi for every i isin N Cmn isin Amn Bmn

In the following example we show what we mean by the hyperoperation Notice that thehyperoperation will be later on used to construct a hypergroup in Theorem 1 which will fulfil theassumption regarding input alphabet stated in Definition 2

Example 2 Consider two elements [~a A23] [~c C23] isin I2VM where [~a A23] =[

(2 62 0 1 4 minusπ8 )

[1 minus1 11 minus1 minus1

]]and [~c C23] =

[(2 81 1 2 5 π

6 )

[minus1 minus1 minus11 minus1 minus1

]] Then the

hyperoperarion

[~a A23] lowastp [~c C23] =

[(2 62 0 1 4 minusπ

8)


]]lowastp

[(2 81 1 2 5

π

6)


]]=

[(2 r2 r3 r4 r5 r6) D23] | 62 le r2 le 81 0 le r3 le 1 1 le r4 le 2 4 le r5 le 5minusπ

8le r6 le

π

6

where [~r D23] is defined by Equation (14) ie D23 isin[

1 minus1 11 minus1 minus1

]


]

164


Theorem 1 For every index p isin N the pairs (IpVM lowastp) are hypergroups

Proof First we have to show that the associativity axiom holds ie for all[~u Amn] [~v Bmn] [~w Cmn] isin Ip

VM there is [~u Amn] lowastp([~v Bmn] lowastp [~w Cmn]

)=(

[~u Amn] lowastp [~v Bmn])lowastp [~w Cmn] Without losing generality we can show that associativity

axiom holds for both matrices and vectors First we consider vectors

~u lowastp(~v lowastp ~w

)= ~u lowastp ~r minvi wi le ri le maxvi wi =

⋃

~sisin~rminvi wilerilemaxvi wi~u lowastp~s =

⋃

~sisin~rminvi wilerilemaxvi wi

~t minui si le ti le maxui si

=

~t minui minvi wi le ti le maxui maxvi wi

=~t minui vi wi le ti le maxui vi wi

=

~t minminui vi wi le ti le maxminui vi wi

=

⋃

~aisin~bminui vilebilemaxui vi~c minwi ai le ci le maxwi ai =

⋃

~aisin~bminui vilebilemaxui vi~a lowastp ~w =

~b minui vi le bi le maxui vi

lowastp ~w =

(~u lowastp ~v

)lowastp ~w

In the case of matrices the proof is straightforward

A lowastp(B lowastp C

)= A lowastp B C = A lowastp B cupA lowastp C = A B cup A C = A B C

= A C cup B C = A lowastp C cup B lowastp C = A B lowastp C =(A lowastp B

)lowastp C

The reproductivity axiom holds automatically because the hyperoperation defined above isobviously extensive ie for all A B isin Ip

VM there is A B sube A lowastp B So for arbitrary [~v Bmn] we have[~v Bmn] lowast Ip

VM =⋃

[~uCmn ]isinIpVM

[~v Bmn] lowast [~u Cmn] = IpVM Thus for all indices p isin N the structures

(IpVM lowastp) are hypergroups

At this point everything is ready for the construction of an e-quasi-multiautomaton with the stateset Sp

RM and input set IpVM for p-th quasi-automaton where p isin N

Remark 2 We consider ordered pairs of vectors and matrices as elements of the input set and also of the stateset As a result we will denote elements with index ı for input and with index s for the state ie

[~aı ı

mn

]A is

an input word and[us Ms

mn]

is a state

Theorem 2 For every index p isin N the triple MAp = (IpVMSp

RM δp) is an e-quasi-multiautomtaton withinput hypergroup (Ip

VM lowastp) where transition function δp IpVM times Sp

RM rarr SpRM is defined by

δp

([~aı ı

mn

]C[vs Ks

mn])

= (15)

(radic

a1 middot v1 a2 middot v2 + a2 a3 middot v3 a4 + v4 equiv mod 2l minus 1 a5 middot v5 a6 + 0 middot v6)s

c11k11 c1nk1n

cm1km1 cmnkmn

s

Proof This proof is constructed as follows first we calculate left hand side condition E-GMACsecond we show that left hand side is included in right hand side

165


The left hand side

δp

([~aı ı

mn

]A δp

([~bı Bı

mn

][us Us

mn]))

=

δp

([~aı ı

mn

]A[(radic

b1u1 b2u2 + b2 b3u3 b4 + u4 equiv mod 2l minus 1 b5u5 b6 + 0 middot u6

)s

(bijuij)s])

=

δp

([~aı ı

mn

]A[(radic

p2 b2u2 + b2 b3u3 b4 + u4 equiv mod 2l minus 1 b5u5 b6

)

s (bijuij)

s])

=

[(radic

a1 p a2(b2u2 + b2) + a2 a3b3u3 a4 + b4 + u4 equiv mod 2l minus 1 a5b5u5 a6 + (0 middot b6))s (aijbijuij)s] =

[(radicp2 a2(b2u2 + b2) + a2 a3b3u3 a4 + b4 + u4 equiv mod 2l minus 1 a5b5u5 a6

)

s (aijbijuij)

s]=

[(p a2(b2u2 + b2) + a2 a3b3u3 a4 + b4 + u4 equiv mod 2l minus 1 a5b5u5 a6)s (aijbijuij)

s] =[vs Vs

mn]

The right hand side

δp

([~aı ı

mn

]A lowastp

[~bı Bı

mn

][us Us

mn])cup δp

(IpVM

[us Us

mn])

From the definition of hyperoproduct lowastp in Equation (14) we simply observe that the left handside is not included in the right part of the right hand side union Therefore we have to proof that theleft hand side belongs to the left part of the right hand side union Therefore we calculate

δp

(IpVM

[us Us

mn])

=⋃

[~pı Pımn]isinIp

VM

δp

([~pı Pı

mn

][us Us

mn])

There exist an input word[~cı Cı

mn

]isin Ip

VM such that the vector is ~c =

(p a2i2s+a2i2+a

s+1 a3b3 a4 + b4 a5b5 a6

)and the matrix is C =

a11b11 a1nb1n

am1bm1 amnbmn

i

Thus

[vs Vs

mn]= δp

([~cı Cı

mn

][us Us

mn])isin

⋃

[~pı Pımn]isinIp

VM

δp

([~pı Pı

mn

][us Us

mn])

As a result E-GMAC holds and the structure MAp = (IpVM Sp

VM δp) is ane-quasi-multiautomaton

Once Theorem 2 is proved we will show a practical application af e-quasi-multiautomata inintelligent transport systems In the following example the e-quasi-multiautomaton represents anautonomous vehicle Its state is described by parameters organized into a sextuple while the matrix isused to detect its environment This example is also linked to Example 1

Example 3 We consider matrix B equivalent to matrix A of Equation (12) which describes the situation inFigure 3 The vehicle A3mdashin Figure 4 depicted in green (while other non-autonomous vehicles are red)mdashdetectsits surroundings and saves the data to matrix B This model is suitable for a situation where only one vehiclesuch as A3 is autonomous and it is not possible to establish communication with other vehicles (because they arenot autonomous) We consider a state

[as Bs

mn]isin S3

VM where~as = (3 30 200 2 0 0) and values in matrix B

corresponding to values in the first picture of Figure 4 If we apply the input[~sı Sı

mn

]isin I3VM by the transition

166


function δ3 where ~sı = (3 1 0 99 1 0minusπ4 ) and Sı

mn =

1 1 1 1 1 1 minus1 1 1minus1 1 1 minus1 1 1 minus1 1 1minus1 1 1 minus1 1 1 1 1 1minus1 1 1 minus1 1 1 1 1 1

1 1 1 1 1 1 1 1 11 1 1 1 1 1 minus1 1 11 1 1 1 1 1 minus1 1 11 1 1 1 1 1 minus1 1 11 1 1 1 1 1 minus1 1 11 1 1 1 1 1 minus1 1 11 1 1 1 1 1 minus1 1 1

then the

first component of the input word is used to control the direction ie it changes the state of the sextupleThe second component of the input word changes the matrix that detects the surroundings of a vehicle This waywe obtain a new state

[rs Rs

mn] where~si = (3 30 198 3 0minusπ

4 ) and the entries of the matrix Rsmn are minus1 for

a21 a24 a31 a34 a41 a44 a17 a27 a67 a77 a87 a97 a107 and 1 for other casesWe know from the new state

[rs Rs

mn]

that the vehicle A3 does not change its velocity or its accelerationWe also know that the vehicle A3 is positioned about 2m far from the intersection border and is between lane 2and 3 and it turns to the left We can see the position surroundings in Figure 4 Consequently this is what weuse in the next step as input for full inclusion in lane 3

Lane 3 Lane 2 Lane 1 4 Lane 3 Lane 2 Lane Lane 4 Lane 3 Lane 2

-1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 1 1 1 -1 -1 -1

-1 -1 -1 1 1 1 -1 -1 -1

-1 -1 -1 1 1 1 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 1 1 1 -1 -1 -1

-1 -1 1 1 1 -1 -1 -1

-1 -1 1 1 1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 -1 -1 -1 -1 -1 1 1

-1 -1 -1 -1 -1 -1 -1 1 1

-1 -1 -1 -1 -1 -1 -1 1 1

-1 1 1 1 -1 -1 -1

-1 1 1 1 -1 -1 -1

-1 1 1 1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 1 1-1 -1 -1 -1 -1 -1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1 -1 -1 -1 -1 -1

1 1 1

1 1 11 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1

0 0 0

0 0 0

-1 -1 -1 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 1

-1 -1 -1 0 0 0 -1 -1 1

-1 -1 -1 0 0 0 -1 1 1

-1 -1 -1 0 0 0 -1 1 1

1

-1-1

-1-1

-1 -1

Lane 3 LaLane 2ne 4

Figure 4 Lane change

We will use the e-quasi-multiautomata from Theorem 2 for the construction of a system calledSMAil Necessarily a change on any component of the state of the system must trigger a changeon another component of the state However there may be situations where the change of othercomponents are not necessarily required Therefore we include the following theorem in which weassert that there are such inputs for which the state of the multiautomata do not change

Theorem 3 For every e-quasi-mulltiautomaton MAp = (IpVMSp

RM δp) there is an input[~eı Eı

mn

]for

which holds

δp

([~eı Eı

mn

][ss Ss

mn])

=[ss Ss

mn]

(16)

167


Proof Consider an arbitrary state[ss Ss

mn]isin SpRM of the e-quasi-mulltiautomaton MAp For input

words[~eı Eı

mn

]=

(e1 e2 e3 e4 e5 e6)

111 11n 1m1 1mn

i where e1 = p e2 = s2

s2+1 e3 = 1 e4 =

0 e5 = 1 e6 = s6 there holds Equation (16) Indeed

δp

([~eı Eı

mn

][ss Ss

mn])

=

δp

(p

s2

s2 + 1 1 0 1 s6)i

111 11n 1m1 1mn

i

(s1 s2 s3 s4 s5 s6)s

a11 a1n am1 amn

s

=

(radic

p2s2

s2 + 1s2 +

s2

s2 + 1 1 middot s3 0 + s4 equiv mod kminus 1 1 middot s5 s6 + 0 middot s6

)

111a11 11na1n

1m1am1 1mnamn

s =

(

ps2(s2 + 1)

s2 + 1s3 s4 s5 s6

)

a11 a1n am1 amn

s =

[ss Ss

mn]

In Example 4 we consider different e-quasi-multiautomata as parts of a cooperative intelligenttransport system Some e-quasi-multiautomata represent autonomous vehicles while some arenon-autonomous vehicles Obviously non-autonomous vehicles cannot detect their surroundingsIn such cases we will use e-quasi-multiautomata in which all entries of state matrices are 0This explains the inclusion of the following lemma

Lemma 1 Let SpRM =[s A]|s isin Rp

6 aij = 0

then a structure subMAp = (IpVMSpRM δp) is

a sub-e-quasi-multiautomaton of e-quai-multiautomaton MAp = (IpVMSp

RM δp)

Proof It is obvious that SpRM sub SpRM and δp Ip

VMtimes SpRM rarr SpRM Next we can see that E-GMACholds because all entries in the state matrix are zero Thus the second component of the state cannot change for any input Therefore the proof for the first component is the same as the proof of theTheorem 2

In what follows we will note that every state[

pss pSsmn]

is equal to[(

pss1 pss

6)

pbsij

]isin SpRM

and every input[

p~aıpAımn

]is equal to

[(paı

1 paı6)

paıij

]isin IpVM from e-quasi-multiautomaton

MAp = (IpVMSp

RM δp) where p isin N is an index In the other words such as pss1 the upper index s

denotes state element lower index 1 denotes position in the sextuple and the lower index p denotesthe pminus th e-quasi-multiautomaton The indices in matrices have the same meaning As far as inputwords are regarded we use upper index ı other indices have the same meaning

Now we need to define the matrix of internal links First we define ϕpp SpRM rarr Ip

VM for allp isin 1 2 n by

ϕpp([(pss

1pss2pss

3pss4pss

5pss6)pAs

mn])

=[(pvı

1pvı2pvı

3pvı4pvı

5pvı6)pBı

mn]

where pvı1 = pss

1 = p pvı2 = pss

2

pss2+

1pss

5

pvı3 = 0 95 pvı

4 = pss4 + l + sgn

(ss

6)

pvı5 = 1

pss5 pvı

6 = 0 and

entries of the input matrix are dependent on the occupancy of the tiles

168


Corollary 1 For mapping ϕpp added to every e-quasi-multiautomaton MAp = (IpVMSp

RM δp) fromTheorem 2 there holds the condition E-GMAC

Proof The proof is obvious On the left hand side we have up to four changes of the original statein the E-GMAC condition (two inputs and two applications of the internal link) However we havea suitable input for the right hand side similar to the proof of Theorem 2

Now we will define a mapping for two different e-quasi-multiautomata ie from the stateset of the pth e-quasi-multiautomaton to the input set of the qth e-quasi-multiautomaton We definemapping ϕpq Sp

RM rarr IqVM for all p q isin 1 2 n where p 6= q between two e-quai-multiautomata

MAp = ((IpVMSp

RM δp)) and MAq = ((IqVMSq

RM δq)) by

ϕpq([(pss

1 pss2 pss

3 pss4 pss

5 pss6)pAs

nm])

=[(qvı

1 qvı2 qvı

3 qvı4 qvı

5 qvı6)qBı

nm]

(17)

where

qvı1 = q 6= pss

1 qvı2 =

pss2

pss2 + 1

+ α pss5 + λ middot |pss

6|10

qvı3 = 0 99 qvı

4 = 0q vı5 = 1q vı

6 = 0

Next we add a meaning of the parameters λ and α

λ =

1 if pss4 gtq ss

4 and pss6 gt 0

1 if pss4 ltq ss

4 and pss6 lt 0

0 otherewise

and α =

1 if pss4 =q ss

4 and qss3 ltp ss

4 and pss5 gt 1

0 otherewise

At this point we can give the main theorem of this section in which we are going to use all resultsobtained above ie e-quasi-multiautomata hypergroups and internal links By Definition 6 we obtainthe n-ary cartesian composition of e-quasi-multiautomata with internal links

Theorem 4 Let MA1 = (I1VMS1

RM δ1)MA2 = ((I2VMS2

RM δ2)) MAp = ((IpVMSp

RM δp)) be ane-quasi-multiautomata with disjoint input-sets Ip

VM and p ge 2 Then a quadruple

SMAil =

(( p⋃

i=1

IiVM

)

potimes

i=1

SiRM

p

prodi=1

δi Mnn(ϕ)

)

is a system of the cartesian composition of e-quasi-multiautomata with internal links

Proof We have to demonstrate that the condition E-GMAC holds ie that there is(

p

prodi=1

δi

)([~aı Aı

mn

]

(p

prodi=1

δi

)([~bı Bı

mn

]

potimes

i=1

[iss iS

smn]))isin

(p

prodi=1

δi

)([~aı Aı

mn

]lowastp

[~bı Bı

mn

]

potimes

i=1

[iss iS

smn])cup

potimes

i=1δi

(IiVM

[iss iS

smn])

(18)

We prove while maintaining generality that

(t

prodi=1

δi

)([~aı Aı

mn

]

(t

prodi=1

δi

)([~bı Bı

mn

]

totimes

i=1

[iss iSs

mn]))isin

totimes

i=1

δi

(IiVM

[iss iSs

mn])

(19)

ie Formula (18) without the left part of the right hand side of E-GMAC There are two cases The firstboth input words are in the same input set and the second input words are from different input sets

169


(a) For the first case we consider[

p~aıpAımn

][

p~bıpBımn

]isin Ip

VM On the left hand of

Equation (19) we know that the input[

p~bıpBımn

]works upon one component from

([1ss 1Ss

mn]

[

2ss 2Ssmn]

[

tss tSsmn])

and the state can alter by the internal link ϕpp This is

performed again for input[

p~aıpAımn

] then we have an input

[p~cıpCı

mn

]isin IpVM for which holds

following (considering the proof of Theorem 2)

δp([

p~aıpAımn

] δp(

[p~bıpBı

mn

][

pss pSsmn])) = δp(

[p~cıpCı

mn

][

pss pSsmn])

When the inputs[

p~aıpAımn

]or[

p~bıpBımn

]are applied on the state

[tss tSs

mn]

other states reactto this change by mapping ϕpq It is obvious that mapping ϕpr influences other states by the inputfrom the respective input set Then for every component of the state on the right side there existsan element from the corresponding input set that state on the left hand side is included on theright hand side

(b) For the second case we consider different inputs[

p~aıpAımn

]isin Ip

VM[

q~bıqBımn

]isin Iq

VMnow On the left-hand side we have an influence on two different components in the tuple([

1ss 1Ssmn]

[

2ss 2Ssmn]

[

tss tSsmn])

it is evident that the same inputs are the same on theright-hand side if internal links ϕpp ϕqq do not change At the moment ϕpp ϕpp influencecorresponding components of the state we have suitable inputs on the right-hand side as inproof of Theorem 2 For mapping the influence of ϕpr and ϕqs on other components we have thesame situation as case a) Thus it is obvious that SMAil holds condition E-GMAC

In the conclusion of this section we will demonstrate the theory explained by using the exampleof SMAil to describe and model a situation with several autonomous vehicles in traffic lanes eachintending to perform some action

Example 4 We will consider five e-quasi-multiautomata MA1 = (I1VMS1

RM δ1)MA2 =

(I2VMS2

RM δ2)MA3 = (I3VMS3



RM δ5) where everye-quasi-multiautomaton represents a vehicle The e-quasi-multiautomata MA2MA3MA4 are autonomousvehicles and MA1MA5 are ordinary vehicles Figure 5 depicts the state of every vehicle where the vehiclesMA1MA5 are denoted in red colour in the same as verge of the road and other colour are used for autonomousvehicles The complete situation with detection field of each vehicle is presented in Figure 6 In fact Figure 5 isstate of SMAil ie

([1rs 1Rs

mn]

[

2ss 2Ssmn]

[

3ts 3Tsmn]

[

4us 4Usmn]

[

5vs 5Vsmn])

Next we need a matrix of the internal link

0 0 0 0 00 ϕ ϕ ϕ 00 ϕ ϕ ϕ 00 ϕ ϕ ϕ 00 0 0 0 0

where 0 means no internal link and ϕ mean internal link between vehicles given by the respective indicesWe are going to describe how to proceed ie what input symbols to use if the grey car represented bye-quasi-multiautomaton MA3 wants to change lane and turn left to lane 5

We have two approaches to start changing the lane we can correct of the state of the vehicle directly by thecomponent on the 2nd 3rd and 4th position in the input word or to use the internal link and 6th component ofthe input We will use an input

[(33ı 31ı 3098ı 30ı 31ı 3

π5

ı) 3(bij)ı= 1

]isin I3VM which will operate on

170


the 6th component Then the resulting statemdashfull correctionmdashwill be made by the internal link The calculationwill be done according to Equations (7) and (15) ie

δ3

([(33ı 31ı 3099ı 30ı 31ı 3

π

5

ı) 3(bij)ı= 1

][

3ts 3Tsmn])

=[(

33 330 319206 33 31 3π

5

)3Ts

nm

](20)

We obtain a state with new elements on the 5th 6th positions where these components have an effect onother components by means of the internal link ϕ33 between the state and input of the e-quasi-multiautomatonMA3 With this internal link we obtain a new input

ϕ33

([(33s 330s 319206s 33s 31s 3

π

5

s)3Ts

nm

])=[(33ı 31ı 31ı 31ı 31ı 30s) 3Cı

nm]

where 3Cınm =

1 1 1 1 1 1 1 1 1 1 minus1minus1 1 1 minus1 1 1 1 1 1 1 minus1minus1 1 1 minus1 1 1 1 1 1 1 minus1minus1 1 1 minus1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 11 1 1 1 minus1 1 1 minus1 1 1 minus11 1 1 1 minus1 1 1 minus1 1 1 minus11 1 1 1 minus1 1 1 minus1 1 1 minus1

ie matrix to correct the state matrix

detecting other vehicles or obstacles near vehicle A3When we apply a new input on the state

[(33 330 319206 33 31 3

π5)

3Tsnm] we obtain

δ3([(33ı 31ı 31ı 31ı 31ı 30ı) 3Cs

nm]

[

3ts 3Tsmn])

=[(33s 330s 319206s 34s 31s 30s) 3Ds

nm]

where matrix D has the same size and entries as depicted for MA3 in Figure 7Now consider internal link ϕ32 ie state of the e-quasi-multiautomaton MA2 will be changed by the state[(

33s 330s 319206s 33s 31s 3π5

s) 3Tsnm]

of the e-quasi-multiautomaton MA3 which we obtained inEquation (20)

We will proceed using the definition of the link between two different e-quasi-multiautomata givenas Equation (17)

ϕ32

([(33s 330s 319206s 33s 31s 3

π

5

s)3Ts

nm

])=

[(22ı 2

(3031

+ 00628)ı

2099ı 20ı 21ı 20ı)

2Mınm

] (21)

where 2Mınm is a suitable matrix which enables us to obtain a state matrix with entries given in Figure 7 for

the state of MA2 After we apply the input obtained by the internal link ϕ32 in Equation (21) with the help oftransition function δ2 we get a new state depicted in Figure 7 for e-quasi-multiautomaton MA2

Next state of the (vehicle) MA4 will not affect velocity by internal link ϕ24 because the input obtained byϕ24 from state

[(33s 330s 319206s 33s 31s 3

π5

s) 3Tsnm]

has parameters λ = 0 and α = 0

Thus the input[(

44ı 3( 30

30+1)ı

3099ı 30ı 31ı 30s)

3Nımn

]operates as neutral input on states of MA4

except for the distance given by 4ss3 Thus we will present a new situation on the lanes ie a new state after the

application of one input See Figure 7

171


Figure 5 SMAil for lane change

Figure 6 Complete situation in lanes with detection field of each vehicle

Figure 7 A new state of SMAil with the release position

4 Conclusions

In this paper we presented the concept of cartesian compositions which was first introduced byW Doumlrfler In the past cartesian compositions were generalized in the sense of hyperstructure theory(using complete hypergroups) Now we considered the internal link between the cartesian productof state sets We described these internal links by matrices and decision functions These functionsdetermine which state will influence other components Our modified concept of the cartesiancomposition is suitable to describe and control systems used for real-life applications as was shown inexamples throughout the paper While constructing our system called SMAil we assumed that it wasmade up of e-quasi-multiautomata of a similar nature This fact affected the proof of the E-GMACcondition (see Theorem 4) which was a substantial simplification of the procedure discussed in [13]In our future research we shall concentrate on answering the question of whether the internal linkmay not remove the necessity to use the extensions of GMAC ie whether the pure GMAC could beused instead of E-GMAC

172


Funding This article was produced with the financial support of the Ministry of Education Youth andSports within the National Sustainability Programme I project of Transport RampD Centre (LO1610) on theresearch infrastructure acquired from the Operation Programme Research and Development for Innovations(CZ1052100030064)

Conflicts of Interest The author declares no conflict of interest

References

1 Corsini P Leoreanu V Applications of Hyperstructure Theory Kluwer Academic Publishers DodrechtThe Netherlands Boston MA USA London UK 2003

2 Davvaz B LeoreanundashFotea V Applications of Hyperring Theory International Academic Press Palm HarborFL USA 2007

3 Chvalina J Smetana B Series of Semihypergroups of Time-Varying Artificial Neurons and RelatedHyperstructures Symmetry 2019 11 927 [CrossRef]

4 Cristea I Kocijan J Novaacutek M Introduction to Dependence Relations and Their Links to AlgebraicHyperstructures Mathematics 20197 885 [CrossRef]

5 Krehliacutek Š Vyroubalovaacute J The Symmetry of Lower and Upper Approximations Determined by a CyclicHypergroup Applicable in Control Theory Symmetry 2020 12 54 [CrossRef]

6 Novaacutek M Krehliacutek Š ELndashhyperstructures revisited Soft Comput 2018 22 7269ndash7280 [CrossRef]7 Al Tahan M Hoskova-Mayerova Š Davvaz B Some Results on (Generalized) Fuzzy Multi-Hv-Ideals of

Hv-Rings Symmetry 2019 11 1376 [CrossRef]8 Chvalina J Functional Graphs Quasi-Ordered Sets and Commutative Hypergroups Masaryk University Brno

Czech Republic 1995 (In Czech)9 Doumlrfler W The cartesian composition of automata Math Syst Theory 1978 11 239ndash257 [CrossRef]10 Geacutecseg F Peaacutek I Algebraic Theory of Automata Akadeacutemia Kiadoacute Budapest Hungary 197211 Ginzburg A Algebraic Theory of Automata Academic Press New York NY USA 196812 Novaacutek M Krehliacutek Š Stanek D n-ary Cartesian composition of automata Soft Comput 2019 24 1837ndash1849

[CrossRef]13 Chvalina J Krehliacutek Š Novaacutek M Cartesian composition and the problem of generalising the MAC

condition to quasi-multiautomata An Univ Ovidius Constanta-Ser Mat 2016 24 79ndash10014 Ashrafi AR Madanshekaf A Generalized action of a hypergroup on a set Ital J Pure Appl Math 1998 15

127ndash13515 Chvalina J Infinite multiautomata with phase hypergroups of various operators In Proceedings of the 10th

International Congress on Algebraic Hyperstructures and Applications Brno Czech Republic 3 September2008 Hoškovaacute Š Ed University of Defense Washington DC USA pp 57ndash69

16 Chvalina J Chvalinovaacute L State hypergroups of automata Acta Math Inform Univ Ostrav 1996 4 105ndash12017 Chvalina J Hoškovaacute-Mayerovaacute Š Dehghan Nezhad A General actions of hyperstructures and some

applications An Univ Ovidius Constanta-Ser Mat 2013 21 59ndash82 [CrossRef]18 Massouros GG Hypercompositional structures in the theory of languages and automata An St Univ AI

Ccediluza Iasi Sect Inform 1994 3 65ndash7319 Doumlrfler W The direct product of automata and quasi-automata In Mathematical Foundations of Computer

Science 5th Symposium Mazurkiewicz A Ed Springer Gdansk Poland 1976 pp 6ndash1020 Chvalina J Novaacutek M Krehliacutek Š Hyperstructure generalizations of quasi-automata induced by modelling

functions and signal processing In Proceedings of the 16th International Conference of Numerical Analysisand Applied Mathematics Rhodes Greece 13ndash18 September 2018

21 Krehliacutek Š Novaacutek M Modifed Product of Automata as a Better Tool for Description of Real-Life SystemsIn Proceedings of the 17th International Conference of Numerical Analysis and Applied MathematicsRhodes Greece 23ndash29 September 2019

22 Golestan K Seifzadeh S Kamel M Karray F Sattar F Vehicle Localization in VANETs Using Data Fusionand V2V Communication In Proceedings of the Second ACM International Symposium on DesignandAnalysis of Intelligent Vehicular Networks and Applications Paphos Cyprus 21 October 2012 pp 123ndash130

23 Novaacutek N Krehliacutek Š Ovaliadis K Elements of hyperstructure theory in UWSN design and dataaggregation Symmetry 2019 11 734 [CrossRef]

173


24 Dresner K Stone P A multiagent approach to autonomous intersection management J Artif Intell Res2008 31 591ndash656 [CrossRef]

25 Fajardo D Au T-C Waller ST Stone P Yang D Automated Intersection Control Performance of FutureInnovation Versus Current Traffic Signal Control Transp Res Rec 2011 2259 223ndash232 [CrossRef]

26 Huifu J Jia H Shi A Meng W Byungkyu BP Eco approaching at an isolated signalized intersectionunder partially connected and automated vehicles environment Transp Res Part C Emerg Technol 2017 79290ndash307

27 Latombe J Robot Motion Planning Kluwer Academic Publishers Boston MA USA 199128 Liu F Naraynan A Bai Q Effective methods for generating collision free paths for multiple robots based

on collision type (demonstration) In Proceedings of the 11th International Conference on AutonomousAgents and Multiagent Systems Valencia Spain 4ndash8 June 2012 Volume 3 pp 1459ndash1460

copy 2020 by the author Licensee MDPI Basel Switzerland This article is an open accessarticle distributed under the terms and conditions of the Creative Commons Attribution(CC BY) license (httpcreativecommonsorglicensesby40)

174

mathematics

Article

Derived Hyperstructures from Hyperconics

Vahid Vahedi 1 Morteza Jafarpour 1dagger Sarka Hoskova-Mayerova 2 Hossein Aghabozorgi 1 Violeta Leoreanu-Fotea 3 and Svajone Bekesiene 4

1 Department of Mathematics Vali-e-Asr University of Rafsanjan Rafsanjan 7718897111 Iranvvahedivruacir (VV) haghabozorgivruacir (HA)

2 Department of Mathematics and Physics University of Defence in Brno Kounicova 65 662 10 BrnoCzech Republic sarkamayerovaunobcz

3 Department of Mathematics Al I Cuza University 6600 Iashi Romania violetafoteauaicro4 Departament of Defence Technologies The General Jonas Žemaitis Military Academy of Lithuania Silo g5

10322 Vilnius Lithuania svajonebekesienelkalt Correspondence mjvruacirdagger The corresponding author in deep gratitude dedicates this work and gives special thanks to Cardiovascular

Surgeon A R Alizadeh Ghavidel

Received 26 January 2020 Accepted 11 March 2020 Published 16 March 2020

Abstract In this paper we introduce generalized quadratic forms and hyperconics over quotienthyperfields as a generalization of the notion of conics on fields Conic curves utilized in cryptosystemsin fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curvegroups We associate some hyperoperations to hyperconics and investigate their properties At theend a collection of canonical hypergroups connected to hyperconics is proposed

Keywords hypergroup hyperring hyperfield (hyper)conics quadratic forms

MSC 20N20 14H52 11G05

1 Introduction

In 1934 Marty initiated the notion of hypergroups as a generalization of groups and referred toits utility in solving some problems of groups algebraic functions and rational fractions [1] To reviewthis theory one can study the books of Corsini [2] Davvaz and Leoreanu-Fotea [3] Corsini andLeoreanu [4] Vougiouklis [5] and in papers of Hoskova and Chvalina [6] and Hoskova-Mayerova andAntampoufis [7] In recent years the connection of hyperstructures theory with various fields hasbeen entered into a new phase For this we advise the researchers to see the following papers (i) Forconnecting it to number theory incidence geometry and geometry in characteristic one [8ndash10] (ii) Forconnecting it to tropical geometry quadratic forms [1112] and real algebraic geometry [1314] (iii) Forrelating it to some other objects see [15ndash19] M Krasner introduced the concept of the hyperfield andhyperring in Algebra [2021] The theory which was developed for the hyperrings is generalizingand extending the ring theory [22ndash25] There are different types of hyperrings [222526] In the mostgeneral case a triplet (R+ middot) is a hyperring if (R+) is a hypergroup (R middot) is a semihypergroup andthe multiplication is bilaterally distributive with regards to the addition [3] If (R middot) is a semigroupinstead of semihypergroup then the hyperring is called additive A special type of additive hyperringis the Krasnerrsquos hyperring and hyperfield [2021242728] The construction of different classes ofhyperrings can be found in [29ndash33] There are different kinds of curves that basically are used incryptography [3435] An elliptical curve is a curve of the form y2 = p(x) where p(x) is a cubicpolynomial with no-repeat roots over the field F This kind of curves are considered and extendedover Krasnerrsquos hyperfields in [13] Now let g(x y) = ax2 + bxy + cy2 + dx + ey + f isin F[x y] and

175


g(x y) = 0 be the quadratic equation of two variables in field of F if a = c = 0 and b 6= 0 thenthe equation g(x y) = 0 is called homographic transformation In [14] Vahedi et al extended thisparticular quadratic equation on Krasnerrsquos quotient hyperfield F

G The motivation of this paper goesin the same direction of [14] If in the general form of the equation of quadratic form one supposethat ae 6= 0 and b = 0 then initiate an important quadratic equation which is called a conic Noticethat the conditions which are considered for the coefficients of the equations of a conic curve and ahomographic curve are completely different Until now the study of conic curves has been on fieldsAt the recent works the authors have investigated some main classes of curves elliptic curves andhomographics over Krasnerrsquos hyperfields (see [1314]) In the present work we study the conic curvesover some quotients of Krasnerrsquos hyperfields

2 Preliminaries

In the following we recall some basic notions of Pell conics and hyperstructures theory thatthese topics can be found in the books [23637] Moreover we fix here the notations that are used inthis paper

21 Conics

According to [36] a conic is a plane affine curve of degree 2 Irreducible conics C come in threetypes we say that C is a hyperbola a parabola or an ellipse according as the number of points atinfinity on (the projective closure of) C equals 2 1 or 0 Over an algebraically closed field everyirreducible conic is a hyperbola Let d be a square free integer nonequal to 1 and put

∆ =

d if d equiv 1 (mod 4)

4d if d equiv 2 3 (mod 4)

The conic C Q0(x y) = 1 associated to the principal quadratic form of discriminant ∆

Q0(x y) =

x2 + xy + 1minusd4 y2 if d equiv 1 (mod 4)

x2 minus dy2 if d equiv 2 3 (mod 4)

is called the Pell conic of discriminant Pell conics are irreducible nonsingular affine curves with adistinguished integral point N = (1 0) The problem corresponding to the determination of E(Q) isfinding the integral points on a Pell conic The idea that certain sets of points on curves can be given agroup structure is relatively modern For elliptic curves the group structure became well known onlyin the 1920s implicitly it can be found in the work of Clebsch and Juel in a rarely cited article wrotedown the group law for elliptic curves defined over R and C at the end of the 19th century The grouplaw on Pell conics defined over a field F For two rational points p q isin Q(F) draw the line through Oparallel to line p q and denote its second point of intersection with p lowast q which is the sum of two p qwhere O is an arbitrary point in pell conic perchance in infinity is identity element of group In theFigure 1 the operation is picturised on the conic section QR( f11)

176


O

xi bull11

xj

xi

xj

xi bull11

xj

QR ( f11)

Figure 1 Conic section QR( f11)

Example 1 Consider f11(x) = xminus1 + x over finite field F = Z7 Then we have a Caley table of points(Table 1)

Table 1 Conic group(QZ7 ( f11) bullab

)

bull11 (0 infin) (1 2) (2minus1) (3 1) (minus3minus1) (minus2 1) (minus1minus2) (infin infin)(0 infin) (0 infin) (1 2) (2minus1) (3 1) (minus3minus1) (minus2 1) (minus1minus2) (infin infin)(1 2) (1 2) (minus1minus2) (minus3minus1) (minus2 1) (3 1) (2minus1) (0 infin) (infin infin)(2minus1) (2minus1) (minus3minus1) (1 2) (minus1minus2) (infin infin) (0 infin) (minus2 1) (3 1)(3 1) (3 1) (minus2 1) (minus1minus2) (1 2) (0 infin) (infin infin) (minus3minus1) (2minus1)

(minus3minus1) (minus3minus1) (3 1) (infin infin) (0 infin) (minus1minus2) (1 2) (2minus1) (minus2 1)(minus2 1) (minus2 1) (2minus1) (0 infin) (infin infin) (1 2) (minus1minus2) (3 1) (minus3minus1)(minus1minus2) (minus1minus2) (0 infin) (minus2 1) (minus3minus1) (2minus1) (3+1) (infin infin) (1 2)(infin infin) (infin infin) (minus1minus2) (3 1) (2minus1) (minus2+1) (minus3minus1) (1 2) (0 infin)

The associativity of the group law is induced from a special case of Pascalrsquos Theorem In thefollowing we recall Pascalrsquos Theorem which is a very special case of Bezoutrsquos Theorem

Theorem 1 ([38] Pascalrsquos Theorem) For any conic and any six points p1 p2 p6 on it the opposite sides ofthe resulting hexagram extended if necessary intersect at points lying on some straight line More specifically letL(p q) denote the line through the points p and q Then the points L(p1 p2)cap L(p4 p5) L(p2 p3)cap L(p5 p6)and L(p3 p4) cap L(p6 p1) lie on a straight line called the Pascal line of the hexagon Figure 2

p3

Conicp6

p4

p5

p2

p1

Figure 2 Pascal line of the hexagon

22 Krasnerrsquos Hyperrings and Hyperfields

Let H be a non-empty set and Plowast(H) denotes the set of all non-empty subsets of H Any functionmiddot from the cartesian product H times H into Plowast(H) is called a hyperoperation on H The image of the

177


pair (a b) isin H times H under the hyperoperation middot in Plowast(H) is denoted by a middot b The hyperoperationcan be extended in a natural way to subsets of H as follows for non-empty subsets A B of H defineA middot B =

⋃aisinAbisinB

a middot b The notation a middot A is applied for a middot A and also A middot a for A middot a Generally

we mean Hk = H times Htimes H (k times) for all k isin N and also the singleton a is identified withits element a The hyperstructure (H middot) is called a semihypergroup if x middot (y middot z) = (x middot y) middot z for allx y z isin H which means that ⋃

uisinxmiddotyu middot z =

⋃

visinymiddotzx middot v

A semihypergroup (H middot) is called a hypergroup if the reproduction law holds x middot H = H middot x = Hfor all x isin H

Definition 1 [2] Let (H middot) be a hypergroup and K be a non-empty subset of H We say that (K middot) is asubhypergroup of H and it denotes K 6 H if for all x isin K we have K middot x = K = x middot K

Let (H middot) be a hypergroup an element er (resp el) of H is called a right identity (resp leftidentity el) if for all a isin H x isin a middot er (resp a isin el middot a) An element e is called an identity if for all a isin Ha isin a middot e cap e middot a A right identity er (resp left identity el) of H is called a scalar right identity (respscalar left identity) if for all a isin H a = a middot er (a = el middot x) An element e is called a scalar identity if forall a isin H a = a middot e = e middot x An element aprime isin H is called a right inverse (resp left inverse) of a in H ifer isin a middot aprime for some right identities er in H (el isin aprime middot a) An element aprime isin H is called an inverse of a isin Hif e isin aprime middot a cap a middot aprime for some identities in H We denote the set of all right inverses left inverses andinverses of a isin H by ir(a) il(a) and i(a) respectively

Definition 2 [2] A hypergroup (H middot) is called reversible if the following conditions hold

(i) At least H has one identity e(ii) every element x of H has one inverse that is i(x) 6= empty(iv) x isin y middot z implies that y isin x middot zprime and z isin yprime middot x where zprime isin i(z) and yprime isin i(y)

Definition 3 [223] A hypergroup (H+) is called canonical if the following conditions hold

(i) for every x y z isin H x + (y + z) = (x + y) + z(ii) for every x y isin H x + y = y + x

(iii) there exists 0 isin H such that 0 + x = x for every x isin H(iv) for every x isin H there exists a unique element xprime isin H such that 0 isin x + xprime (we shall write minusx for xprime and

we call it the opposite of x)(v) z isin x + y implies y isin zminus x and x isin zminus y

Definition 4 [2] Suppose that (H middot) and (K ) are two hypergroups A function f H rarr K is called ahomomorphism if f (x middot y) sube f (x) f (y) for all x and y in H We say that f is a good homomorphism if for allx and y in H f (x middot y) = f (x) f (y)

The more general hyperstructure that satisfies the ring-like conditions is the hyperring The notionof the hyperring and hyperfield was introduced in Algebra by M Krasner in 1956 [21] Accordingto the current terminology these initial hypercompositional structures are additive hyperrings andhyperfields whose additive part is a canonical hypergroup Nowadays such hypercompositionalstructures are called Krasnerrsquos hyperrings and hyperfields

Definition 5 [20] A Krasnerrsquos hyperring is an algebraic structure (R+ middot) which satisfies thefollowing axioms

(1) (R+) is a canonical hypergroup

178


(2) (R middot) is a semigroup having zero as a bilaterally absorbing element ie x middot 0 = 0 middot x = 0(3) The multiplication is distributive with respect to the hyperoperation +

A Krasnerrsquos hyperring is called commutative if the multiplicative semigroup is a commutativemonoid A Krasnerrsquos hyperring is called a Krasnerrsquos hyperfield if (Rminus 0 middot) is a commutative groupIn [20] Krasner presented a class of hyperrings which is constructed from rings He proved that if Ris a ring and G is a normal subgroup of Rprimes multiplicative semigroup then the multiplicative classesx = xG x isin R form a partition of R He also proved that the product of two such classes as subsetsof R is a class mod G as well while their sum is a union of such classes Next he proved that the setR = R

G of these classes becomes a hyperring when

(i) xGoplus yG = zG|z isin xG + yG and(ii) xG yG = xyG

Moreover he observed that if R is a field then RG is a hyperfield Krasner named these

hypercompositional structures quotient hyperring and quotient hyperfield respectively At the sametime he raised the question if there exist non-quotient hyperrings and hyperfields [20] Massourosin [27] generalized Krasnerrsquos construction using not normal multiplicative subgroups and provedthe existence of non-quotient hyperrings and hyperfields Since the paper deals only with Krasnerrsquoshyperfields we will write simply quotient hyperfields instead of Krasnerrsquos quotient hyperfields

3 Hyperconic

The notion of hyperconics on a quotient hyperfield will be studied in this section By the use ofhyperconic QF( f

AB) we present some hyperoperations as a generalization group operations on fields

We investigate some attributes of the associated hypergroups from the hyperconics and the associatedHv-groups on the hyperconics

Let g(x y) = ax2 + bxy + cy2 + dx + ey + f isin F[x y] and g(x y) = 0 be the quadratic equation oftwo variables in field of F If c = 0 and equation g(x y) = 0 still stay in quadratic and two variables orthe other word c = 0 and (a b) 6= 0 6= (e b) then it can be calculated as an explicit function y in termsof x also with a change of variables can be expressed in the form of Y = AX2 + BX or AXminus1 + BXwhere A B isin F

For this purpose if a e 6= 0 = b set x = X and y = Y minus f eminus1 then Y = AX2 + BX whereA = minusaeminus1 B = minusdeminus1 If b 6= 0 = a set x = X minus ebminus1 and y = Y minus dbminus1 then Y = AXminus1 whereA = edbminus2 minus f bminus1 If b 6= 0 6= a set x = Xminus ebminus1 and y = Y + 2aebminus2αF minus dbminus1 that

αF =

0 if char(F) = 2

1 if char(F) 6= 2

then Y = AXminus1 + BX where A = minusae2bminus3 + edbminus2 minus f bminus1 and B = minusabminus1 Reduced quadraticequation of two variables ax2 + bxy + dx + ey + f = 0 in field of F can be generalized in quotienthyperfield F

Definition 6 Let F be the quotient hyperfield and (A B) isin F2 and f AB(x) be equal to Axminus1 oplus Bxor Ax2 oplus Bx Then the relation y isin f AB(x) is called generalized reduced two variable quadraticequation in F2 Moreover the set Q( f AB F) = (x y) isin F2|y isin f AB(x) is called conic hypersectionand if A 6= 0 Q( f AB F) is named non-degenerate conic hypersection For all a isin A and b isin BQ( fab F) = (x y) isin F2|y = fab(x) is conic section and for a 6= 0 is non-degenerate conic section in whichfab(z) = az2 + bz or azminus1 + bz corresponding to f AB It is also said to Q( f AB F) =

⋃(xy)isinQ( f AB F)

x times y

conic hypersection as a subset of F2 and Q( fab F) = Q( fab F) =(x y)|(x y) isin Q( fab F)

where

179


(x y) = (x y) for all (x y) isin Q( f AB F)

Theorem 2 Using the above notions we have Q( f AB F) =⋃

aisinAbisinBQ( fab F)

Proof Let (x y) isin Q( f AB F) and without losing of generality f (x) = Ax2 + Bx Then

(x y) isin Q( f AB F)lArrrArr y isin Ax2 oplus Bx

lArrrArr y isin Ax2 + Bx

lArrrArr y = ax2 + bx f or some (a b) isin Atimes B

lArrrArr y = agx2 + bgx f or some g isin G

lArrrArr y = aprimex2 + bprimex where aprime = ag bprime = bg

lArrrArr (x y) isin Q( faprime bprime F) f or some (aprime bprime) isin Atimes B

lArrrArr (x y) isin⋃

aisinAbisinB

Q( fab F)

Consequently Q( f AB F) =⋃


Example 2 Let F = Z5 be the field of order 5 G = plusmn1 6 Flowast and f10(x) = x2 Then we have F = 0 1 2

Q( f10

F) = Q( f10 F) cupQ( f(minus1)0 F) where

Q( f10 F) = (0 0) (1 1) (minus1 1) (2minus1) (minus2minus1)

Q( f(minus1)0 F) = (0 0) (1minus1) (minus1minus1) (2 1) (minus2 1)

and Q( f10 F) = Q( f10 F) = (0 0) (1 1) (2 1) = Q( f(minus1)0 F) = Q( f

(minus1)0 F) In this case Q( f10

F) isa non-degenerate conic hypersection because A = 1 6= 0

Definition 7 Let F be a field x isin F and G be a subgroup in Flowast We take

O =

0minus1 if fab(z) = az2 + bz0 if fab(z) = azminus1 + bz

Gx( fab) =

x if G = 1z isin F| fab(z) = fab(x) if G 6= 1 fab(z) = az2 + bz

minusx x if G 6= 1 fab(z) = azminus1 + bz

Obviously 0minus1 is an element outside of F We denote 0minus1 = 10 by infin where infin isin F and infin = infin

Suppose that Ginfin( fab) = infin fab(infin) = infin fab(O) = infin for all a isin A b isin B also X = x|x isin X

where x = (x fab(x)) and X sube F cup O Moreover O middot O = O = O +O x middot O =

O if x 6= 0

0 if x = 0and

x +O =

O if O = 0minus1

x if O = 0 for all x in field of (F+ middot)

Remark 1 It should be noted that associativity by adding O to field of (F+ middot) for two operations of + andmiddot remains preserved

180


Definition 8 Let Q( f AB F) be a non-degenerate conic hypersection Finfin = F cup infin and

QF( fab) = x x isin Finfin x 6isin L0

QF( f AB) =⋃

aisinAbisinB

QF( fab)

where L0 = (x 0)|x isin FO For all xi xi isin QF( fab)

xi bullab xj = (xi bullab xj fab(xi bullab xj)) in which (xi bullab xj 0) = L0 cap Lab(xixj)

and

Lab(xi xj) =

(x y) isin F2|yminus fab(xi) =fab(xj)minus fab(xi)

(xjminusxi)(xminus xi) xi 6= xjO 6isin xi xj

(x y) isin F2|yminus fab(xi) = f primeab(xi)(xminus xi) xi = xj 6isin O(x y) isin F2|O 6= x isin xi xj cup O xi 6= xjO isin xi xj(O y)|y isin Finfin = F cup infin (xi xj) = (OO)

and fprimeab is meant by formal derivative fabWe denote QF( fab) by QF( fab) and QF( f AB) by QF( f AB) also take O = O = O f (O) =

f (O) = f (O) and (O f (O)) = (O f (O)) = (O f (O)) Moreover O O = O and O oplusO = O

also for all x in hyperfield of (Foplus) xO =

O if x 6= 0

0 if x = 0xoplusO =

O if O = 0minus1

x if O = 0and agree to

Lo cap L(xi xj) = (infin 0) if fab(xi) = fab(xj) In addition say to Lab(xi xj) the line passing from xi xjIntuitively each line passing from (O infin) is called vertical line and every vertical line pass through (O infin) Ois playing an asymptotic extension role for function fab

Remark 2 By adding O to hyperfield of (Foplus) associativity for two hyperoperations of oplus and remains preserved

Suppose that x isin Q( fab F) and x =

x fab(x) = ax2 + bx

xminusx fab(x) = axminus1 + bxHence we the

following proposition

Proposition 1 if |QF( fa1b1) capQF( fa2b2

)| ge 2 then Q( fa1b1 F) = Q( fa2b2

F)

Proof Let x1 x2 sube QF( fa1b1) capQF( fa2b2

) x1 6= x2 and i j = 1 2 Then

yi = ajx2i + bjxi =rArr x1 6= x2 =rArr

a1 = a2 =x2y1 minus x1y2

x21x2 minus x2

2x1

b1 = b2 =minusx2

2y1 + x21y2

x21x2 minus x2

2x1

yi = ajxminus1i + bjxi =rArr x1 6= plusmnx2 =rArr


x2xminus11 minus x1xminus1

2

b1 = b2 =minusxminus1

2 y1 + xminus11 y2


2

Hence Q( fa1b1 F) = Q( fa2b2

F) as we expected

181


Definition 9 Let Q( f AB F) be a non-degenerate conic hypersection then it is named hyperconic and denotedto QF( f AB) if the following implication for all a c isin A and b d isin B holds

QF( fab) capQF( fcd) 6=O fab(x) = ax2 + bxO infin fab(z) = axminus1 + bx

=rArr QF( fab) = QF( fcd)

Proposition 2 Let xi = (xi f (xi)) and xj = (xj f (xj)) belong to QF( fab) then

xi bullab xj =

xi fab(xj)minus xj fab(xi)

fab(xj)minus fab(xi)xi 6= xjO 6isin xi xj

xi minusfab(xi)

f primeab(xi)

xi = xj 6isin O

xi xi 6= O = xj

xj xj 6= O = xi

O (xi xj) = (OO)

Proof The proof is straightforward for the first two cases If fab(xi) = fab(xj) then

xi bullab xj =xi fab(xj)minus xj fab(xi)

fab(xj)minus fab(xi)=


0= infin

(xi bullab xj 0) = Lo cap L(xi xj) = (infin 0) =rArr xi bullab xj = infin

Suppose that (xi xj) isin Q2F( fab) by regarding Definition 8 if xi 6= O = xj then

(xi bullabO 0) = L0 cap Lab(xi O) = (xi 0) =rArr xi bullabO = xi

if xj 6= O = xi then proof is similar to previous manner ultimately if xi = xj = O then

(O bullabO 0) = L0 cap L(O O) = (O 0) =rArr O bullabO = O

Remark 3(QF( fab) bullab

)is a conic group for all (a b) isin Atimes B Notice that bullab is the group operation on

the conic QF( fab)

Example 3 Let F = Z5 the field of order 5 G = plusmn1 6 Flowast and f10(x) = x2 Then we have

F = 0 1 2 QF( f10) = QF( f10) cupQF( f

(minus1)0) where QF( f10) = O (1 1) (minus1 1) (2minus1) (minus2minus1)and QF( f10) = O (1 1) (2 1) QF( f

(minus1)0) = O (1minus1) (minus1minus1) (2 1) (minus2 1) and QF( f(minus1)0)

=

O (1 1) (2 1) in this case QF( f10) is a hyperconic because QF( f10) capQF( f

(minus1)0) = O

Definition 10 We introduce hyperoperation on QF( f AB) as followsLet (x y) (xprime yprime) isin QF( f AB) If (x y) isin QF( fab) and (xprime yprime) isin QF( faprime bprime) for some a aprime isin A and b bprime isin B

(x y) (xprime yprime) =

xi bullabxj|(xi xj) isin Gx( fab)times Gxprime( faprime bprime) if QF( fab) = QF( faprime bprime)

QF( fab) cupQF( faprime bprime) otherwise

Theorem 3(QF( f AB)

)is a hypergroup

Proof Suppose that X Y Z sube QF( f AB)by Bezoutrsquos Theorem (x y) (xprime yprime) sube Plowast(QF( fab))Now let X = (x y) isin QF( fab) Y = (xprime yprime) isin QF( faprime bprime) Z = (xprimeprime yprimeprime) isin QF( faprimeprime bprimeprime) whereJ = (a b) (aprime bprime) (aprimeprime bprimeprime) sube A times B If (x y) = (x1 y1) and (xprime yprime) = (xprime1 yprime1) then x = x1

182


and xprime = xprime1 Because Gx( fab) = Gx1( fab) and Gxprime( fab) = Gxprime1( fab) we have Gx( fab) times Gxprime( fab) =

Gx1( fab)times Gxprime1( fab) thus

z bullabw|(z w) isin Gx( fab)times Gxprime( fab) = z bullabw|(z w) isin Gx1( fab)times Gxprime1( fab)

and that is (x y) (xprime yprime) = (x1 y1) (xprime1 yprime1) consequently is well defined If X = (O infin) orY = (O infin) or Z = (O infin) associativity is evident If this property is not met the following casesmay occur

Case1 If |J| = 1 In this case we have QF( fab) = QF( faprime bprime) = QF( faprimeprime bprimeprime)

[(x y) (xprime yprime)] (xprimeprime yprimeprime) =(xi bullabx

primej)|(xi xprimej) isin Gx( fab)times Gxprime( fab)

(xprimeprime yprimeprime)

=

(xibullabxprimej)bullabx

primeprimek |(xi xprimej xprimeprimek ) isin Gx( fab)times Gxprime( fab)times Gxprimeprime( fab)

Similarly

(x y) [(xprime yprime) (xprimeprime yprimeprime)] =

xibullab(xprimejbullabxprimeprimek )|(xi xprimej xprimeprimek ) isin Gx( fab)times Gxprime( fab)times Gxprimeprime( fab)

On the other hand we have

L(xi xprimej) cap L(xi bullab xprimej O) = (xi bullab xj 0) sube L0

L(xprimej xprimeprimek ) cap L(O xprimej bullab xprimeprimek ) = (xj bullab xk 0) sube L0

Therefore by Pascalrsquos Theorem we have

L(xprimeprimek xi bullab xprimej) cap L(xprimej bullab xprimeprimek xi) sube L0

and in addition((xi bullab xprimej) bullab xprimeprimek 0) = L0 cap L(xi bullab xprimej xprimeprimek )

(xi bullab (xprimej bullab xprimeprimek ) 0) = L0 cap L(xi xprimej bullab xprimeprimek )

L0 cap L(xi bullab xprimej xprimeprimek ) = L(xprimeprimek xi bullab xprimej) cap L( xprimej bullab xprimeprimek xi) = L0 cap L(xi xprimej bullab xprimeprimek )

On the other word(xi bullab xprimej) bullab xprimeprimek = xi bullab (xprimej bullab xprimeprimek )

So

((xi bullab xprimej) bullab xprimeprimek

)=

(xi bullab (xprimej bullab xprimeprimek )

)for all (xi xprimej xprimeprimek ) isin Gx( fab)times Gxprime( fab)times Gxprimeprime( fab)

Case2 If |J| = 2 (i) If QF( fab) = QF( faprime bprime) 6= QF( faprimeprime bprimeprime) We have

[(x y) (xprime yprime)] (xprimeprime yprimeprime) = [z bullabw|(z w) isin Gx( fab)times Gxprime( fab)] (xprimeprime yprimeprime)

=⋃

(uv)isin(xy)(xprime yprime)

(u v) (xprimeprime yprimeprime)

= QF( fab) cupQF( faprimeprime bprimeprime)

183


Otherwise

(x y) [(xprime yprime) (xprimeprime yprimeprime)] = (x y) (QF( faprime bprime) cupQF( faprimeprime bprimeprime)

)

= QF( faprime bprime) capQF( fab) cupQF( faprimeprime bprimeprime)


(ii) If QF( fab) 6= QF( faprime bprime) = QF( faprimeprime bprimeprime) This case similar to (i)

(iii) If QF( fab) = QF( faprimeprime bprimeprime) 6= QF( faprime bprime) We have

[(x y) (xprime yprime)] (xprimeprime yprimeprime) =(QF( fab) cupQF( faprime bprime)

) (xprimeprime yprimeprime)

= QF( fab) cupQF( faprime bprime) cupQF( faprimeprime bprimeprime)

= QF( fab) cupQF( faprime bprime)

On the other hand

(x y) [(xprime yprime) (xprimeprime yprimeprime)] = (x y) (QF( faprime bprime) cupQF( faprimeprime bprimeprime))



Case3 If |J| = 3 In this case we have




On the other hand


)


To prove the validity of reproduction axiom for rdquordquo let us consider two casesCase1 If |Atimes B| = 1 then F = F and QF( f AB) = QF( fab) where a isin A b isin B also (QF( fab) )

is a conic group hence there is nothing to proveCase2 If |Atimes B| gt 1 consider arbitrary element x isin QF( fab) sube QF( f AB) then

x QF( f AB) =(

x ⋃

a 6=iisinAb 6=jisinB

QF( fij)) cup (x QF( fab))

=( ⋃


x QF( fij))cupQF( fab)

=(⋃

iisinAjisinB

QF( fij))cupQF( fab)

= QF( f AB)

Similarly QF( f AB) x = QF( f AB) and reproduction axiom is established Thus(QF( f AB)

)is

a hypergroup

Remark 4 The hyperconic and the associated hypergroup are conic and conic group respectively if G = 1

184


Example 4 Let F = Z5 be the field of order 5 and G = plusmn1 6 Flowast We have F = 0 1 2In addition if we go back to Example 3 then QF( f

10) = QF( f10) cup QF( f

(minus1)0) is hyperconic whereQF( f10) = O (1 1) (minus1 1) (2minus1) (minus2minus1)QF( f

(minus1)0) = O (1minus1) (minus1minus1) (2 1) (minus2 1) QF( f10) = O (1 1) (2 1) Now let H = QF( f10)

and K = QF( f(minus1)0) Then H and K are reversible subhypergroups of QF( f

10) which are defined by the Caley

Tables 2 and 3 respectively

Table 2 Cayle table(QZ5 ( f10)

)

O (1 1) (minus1 1) (2minus1) (minus2minus1)O O (plusmn1 1) (plusmn1 1) (plusmn2minus1) (plusmn2minus1)

(1 1) (plusmn1 1) O (plusmn2minus1) O (plusmn2minus1) (plusmn1 1) (plusmn2minus1) (plusmn1 1) (plusmn2minus1)(minus1 1) (plusmn1 1) O (plusmn2minus1) O (plusmn2minus1) (plusmn1 1) (plusmn2minus1) (plusmn1 1) (plusmn2minus1)(2minus1) (plusmn2minus1) (plusmn1 1) (plusmn2minus1) (plusmn1 1) (plusmn2minus1) (plusmn1 1) O (plusmn1 1) O(minus2minus1) (plusmn2minus1) (plusmn1 1) (plusmn2minus1) (plusmn1 1) (plusmn2minus1) (plusmn1 1) O (plusmn1 1) O

Table 3 Caley table(QZ5 ( f(minus1)0)

)

O (1minus1) (minus1minus1) (2 1) (minus2 1)O O (plusmn1minus1) (plusmn1minus1) (plusmn2 1) (plusmn2 1)

(1minus1) (plusmn1minus1) O (plusmn2 1) O (plusmn2 1) (plusmn1minus1) (plusmn2 1) (plusmn1minus1) (plusmn2 1)(minus1minus1) (plusmn1minus1) O (plusmn2 1) O (plusmn2 1) (plusmn1minus1) (plusmn2 1) (plusmn1minus1) (plusmn2 1)(2 1) (plusmn2 1) (plusmn1minus1) (plusmn2 1) (plusmn1minus1) (plusmn2 1) (plusmn1minus1) O (plusmn1minus1) O(minus2 1) (plusmn2 1) (plusmn1minus1) (plusmn2 1) (plusmn1minus1) (plusmn2 1) (plusmn1minus1) O (plusmn1minus1) O

Proposition 3 H is subhypergroup of QF( f AB) if and only if H =⋃

(ij)isinIsubeAtimesBQF( fij) or H 6 Qij(F)

for some (i j) isin Atimes B

Proof (rArr) Let us assume that H QF( fij) for every (i j) in Atimes B Then in Atimes B exist (i j) 6= (s t)such that HcapQF( fij) 6= empty 6= HcapQF( fst) Now let I=(i j) isin Atimes B|HcapQF( fij) 6= empty thus we haveH sube ⋃

(ij)isinIQF( fij) sube

⋃(st)(ij)isinI

(QF( fij) cap H) (QF( fst) cap H) sube H Accordingly H =⋃

(ij)isinIQF( fij)

(lArr) It is obvious

Proposition 4 Let H be a subhypergroup of QF( f AB) Then H is reversible hypergroup if and only ifH 6 QF( fij) for some (i j) isin Atimes B

Proof (lArr) First we prove that H 6 QF( fij) is a regular reversible hypergroup for all (i j) isin Atimes BLet (x y) and (xprime yprime) are elements in QF( fij)

Case1 If xprime 6isin Gx( fab) then

(xprimeprime yprimeprime) isin (x y) (xprime yprime) =rArr (xprimeprime yprimeprime) = z bullijw f or some(z w) isin Gx( fab)times Gxprime( fab)

=rArr xprimeprime = z bullijw

=rArr z = xprimeprime bullijh where w bullijh = O

=rArr (z fab(z)) = xprimeprime bullijh and h isin Gw( fab) = Gxprime( fab)

=rArr (z fab(z)) isin (xprimeprime fab(xprimeprime)) (xprime fab(xprime))

185


Case2 If xprime isin Gx( fab) then z w isin Gx( fab) and m isin z w = x bullab x n bullab n O where x bullab n =

O then z isin m w and w isin z mCase3 If Y isin infin X = X infin then infin isin Y X and X isin infin Y Notice that infin isin X X for all

X isin QF( fij) (ie every element is one of its inverses)(rArr) Assume that (x y) isin H cap QF( fij) and (xprime yprime) isin H cap QF( fst) in which (i j) 6= (s t) (x y) 6=O 6= (xprime yprime) and (xprimeprime yprimeprime) isin (x y) (xprime yprime)capQF( fij) Then (xprime yprime) isin (z w) (xprimeprime yprimeprime) sube QF( fij) wherez isin Gx( fab) Hence QF( fij) = QF( fst) and this means reversibility conditions do not hold

The class of Hv- groups is more general than the class of hypergroups which is introduced byTh Vougiouklis [39] The hyperstructure (H ) is called an Hvminusgroup if x H = H = H x and alsothe weak associativity condition holds that is x (y z) cap (x y) z 6= empty for all x y z isin H In [1314]the authors have investigated some hyperoperations denoted by and on some main classes ofcurves elliptic curves and homographics over Krasnerrsquos hyperfields In the following we study themon hyperconic Consider the following hyperoperation on the hyperconic (QF( f AB)

(x y)(xprime yprime) = (v w)|(v w) isin (xtimes y) (xprime times yprime)

for all (x y) (xprime yprime) in QF( f AB)

Proposition 5 (QF( f AB) ) is an Hv-group


Proposition 6 If ψAB

QF( f AB) minusrarr QF( f AB) ψAB

(x y) = (x y) then ψAB

is an epimorphismof Hv-groups

Proof The base of the proof is similar to the proof of Proposition 3 in [14]

Example 5 Let G = plusmn1 be a subgroup of Flowast where F = Z5 Consider f11(x) = x2 oplus x on F = 0 1 2

Consequently QF( f11) = O (1 2) (2 1) (2 2) is a hyperconic a calculation gives us the Table 4 ofHv-group

Table 4 Conic Hv-group(QZ5

( f11) )

O (1 2) (2 1) (2 2)O O (1 2) (2 2) (2 1) (2 2) (1 2)

(1 2) (1 2) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2)(2 1) (2 1) (1 2) (2 1) (2 2) QF( f11) (1 2) (2 1) (2 2)(2 2) (1 2) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2)

Let A be a finite set called alphabet and let K be a non-empty subset of A called key-set and alsolet ldquomiddotrdquo be a hyperoperation on A In [40] Berardi etal utilized the hyperoperations with the followingcondition k middot x = k middot yrArr x = y for all (x y) isin A2 and k isin K Let the subhypergroup

(QF( fmn)

)of

(QF( fmn)

)and A = ax|x isin QF( fmn) where ax = i(x) Notice that i(x) = Gx( fab) is the set of

all inverses of x for all x isin QF( fmn) We define the hyperoperation on A as bellow

au av = aw|w isin u v

Theorem 4 (A ) is a canonical hypergroup which satisfies Berardirsquos condition

Proof The proof is similar to the one of Theorem 41 in [13]

186


4 Conclusions

Conic curve cryptography (CCC) is rendering efficient digital signature schemes (CCDLP) Theyhave a high level of security with small keys size Let g(x y) = ax2 + bxy + cy2 + dx + ey + f isin F[x y]and g(x y) = 0 be the quadratic equation of two variables in field of F if a = c = 0 and b 6= 0 thenthe equation g(x y) = 0 is called homographic transformation In [14] Vahedi et al extended thisparticular quadratic equation on the quotient hyperfield F

G Now suppose that ae 6= 0 and b = 0 ing(x y) Then the curve is called a conic The motivation of this paper goes in the same directionof [14] In fact by a similar way the notion of conic on a field extended to hyperconic over a quotienthyperfield hyperfield as picturized in Figure 3 Notice that as one can see the group structures of thesetwo classes of curves have different applications the associated hyperstructures can be different inapplications In the last part of the paper a canonical hypergroup which is assigned by

(QF( fnm)

)

is investigated

HYPERCONIC fAB

Figure 3 Hyperconic QR( f AB)

Author Contributions Conceptualization methodology investigation resources writingmdashoriginal draftwritingmdashreview and editing all authors worked on equally project administration and funding acquisitionSH-M All authors have read and agreed to the published version of the manuscript

Funding Sarka Hoskova-Mayerova was supported within the project for development of basic and appliedresearch developed in the long term by the departments of theoretical and applied bases FMT (Project codeDZRO K-217) supported by the Ministry of Defence in the Czech Republic

Conflicts of Interest Authors declare no conflict of interest The funders had no role in the design of the study inthe collection analysis or interpretation of data in the writing of the manuscript or in the decision to publishthe results

References

1 Marty F Sur Une Generalization de Group In Proceedings of the 8th Congres Des MathematiciensScandinaves Stockholm Sweden 14ndash18 August 1934 pp 45ndash49

2 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Tricesimo Italy 19933 Davvaz B Leoreanu-Fotea V Hyperring Theory and Applications International Academic Press Cambridge

MA USA 20074 Corsini P Leoreanu-Fotea V Applications of Hyperstructure Theory Kluwer Academical Publications

Dordrecht The Netherlands 20035 Vougiouklis T Hyperstructures and Their Representations Hadronic Press Palm Harbor FL USA 19946 Hoskova S Chvalina J A survey of investigations of the Brno research group in the hyperstructure theory

since the last AHA Congress In Proceedings of the AHA 2008 10th International Congress-AlgebraicHyperstructures and Applications University of Defence Brno Czech Republic 3ndash9 September 2008pp 71ndash83 ISBN 978-80-7231-688-5

187


7 Antampoufis N Hoskova-Mayerova S A Brief Survey on the two Different Approaches of FundamentalEquivalence Relations on Hyperstructures Ratio Math 2017 33 47ndash60 doi1023755rmv33i0388

8 Connes A Consani C From monoids to hyperstructures In search of an absolute arithmetic In CasimirForce Casimir Operators and the Riemann Hypothesis Mathematics for Innovation in Industry and Science van DijkG Wakayama M Eds De Gruyter Berlin Germany 2010 pp 147ndash198

9 Connes A Consani C The hyperring of adele classes J Number Theory 2011 131 159ndash19410 Connes A Consani C The universal thickening of the field of real numbers In Proceedings of the

Thirteenth Conference of the Canadian Number Theory Association Ottawa ON Canada 16ndash20 June 201411 Viro O On basic concepts of tropical geometry Proc Steklov Inst Math 2011 273 252ndash28212 Gladki P Marshall M Orderings and signatures of higher level on multirings and hyperfields

J-Theory-Theory Its Appl Algebr Geom Topol 2012 10 489ndash51813 Vahedi V Jafarpour M Aghabozorgi H Cristea I Extension of elliptic curves on Krasner hyperfields

Comm Algebra 2019 47 4806ndash482314 Vahedi V Jafarpour M Cristea I Hyperhomographies on Krasner Hyperfields Symmetry Class Fuzzy

Algebr Hypercompos Struct 2019 11 144215 Freni D Strongly Transitive Geometric Spaces Applications to Hypergroups and Semigroups Theory

Comm Algebra 2004 32 969ndash98816 Izhakian Z Knebusch M Rowen L Layered tropical mathematics J Algebra 2014 416 200ndash27317 Izhakian Z Rowen L Supertropical algebra Adv Math 2010 225 2222ndash228618 Lorscheid O The geometry of blueprints Part i Algebraic background and scheme theory Adv Math 2012

229 1804ndash184619 Lorscheid O Scheme theoretic tropicalization arXiv 2015 arXiv15080794920 Krasner M A class of hyperrings and hyperfields Int J Math Math Sci 1983 6 307ndash31121 Krasner M Approximation des corps value complets de caracteristique p10 par ceux de caracteristique

0 In Colloque d Algebre Superieure Tenu a Bruxelles du 19 au 22 Decembre 1956 Centre Belge de RecherchesMathematiques Etablissements Ceuterick Louvain Belgium Gauthier-Villars Paris France 1957pp 129ndash206

22 Ameri R Eyvazi M Hoskova-Mayerova S Superring of Polynomials over a Hyperring Mathematics 20197 902 doi103390math7100902

23 Mittas JD Hypergroupes canoniques Math Balk 1972 2 165ndash17924 Massouros CG A field theory problem relating to questions in hyperfield theory In Proceedings

of the International Conference on Numerical Analysis and Applied Mathematics Halkidiki Greece19ndash25 September 2011 Volume 1389 pp 1852ndash1855

25 Dakic J Jancic-Rasovic S Cristea I Weak Embeddable Hypernear-Rings Symmetry 2019 11 9626 Massouros CG Massouros GG On Join Hyperrings In Proceedings of the 10th International Congress on

Algebraic Hyperstructures and Applications University of Defence Brno Czech Republic 3ndash9 September2008 pp 203ndash215

27 Massouros CG On the theory of hyperrings and hyperfields Algebra Log 1985 24 728ndash74228 Massouros GG Massouros CG Homomorphic Relations on Hyperringoids and Join Hyperrings

Ratio Math 1999 13 61ndash7029 Ameri R Kordi A Sarka-Mayerova S Multiplicative hyperring of fractions and coprime hyperideals

An Stiintifice Univ Ovidius Constanta Ser Mat 2017 25 5ndash23 doi 101515auom-2017-000130 Jancic-Rasovic S About the hyperring of polynomials Ital J Pure Appl Math 2007 21 223ndash23431 Cristea I Jancic-Rasovic S Operations on fuzzy relations A tool to construct new hyperrings J Mult-Val

Logic Soft Comput 2013 21 183ndash20032 Jancic-Rasovic S Dasic V Some new classes of (mn)-hyperrings Filomat 2012 26 585ndash59633 Cristea I Jancic-Rasovic S Composition Hyperrings An Stiintifice Univ Ovidius Constanta Ser Mat 2013

21 81ndash9434 Flaut C Flaut D Hoskova-Mayerova S Vasile R From Old Ciphers to Modern Communications

Adv Mil Technol 2019 14 79ndash88 doi103849aimt0128135 Saeid AB Flaut C Hoskova-Mayerova S Some connections between BCK-algebras and n-ary block codes

Soft Comput 2017 doi101007s00500-017-2788-z

188


36 Hankerson D Menezes A Vanstone SA Guide to Elliptic Curve Cryptography Springer BerlinHeidelbergGermany 2004

37 Koblitz N Introduction to elliptic curves and modular forms In Volume 97 of Graduate Texts in MathematicsSpringer New York NY USA 1984

38 Silverman JH Tate IT Rational Points on Elliptic Curves Springer New York NY USA 1992 p 28939 Vougiouklis T The fundamental relation in hyperrings The general hyperfield In Proceedings of the

4th International Congress on AHA Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991pp 209ndash217

40 Berardi L Eugeni F Innamorati S Remarks on Hypergroupoids and Cryptography J Combin InformSyst Sci 1992 17 217ndash231


189

mathematics

Article

On the Theory of LeftRight Almost Groups and Hypergroupswith their Relevant Enumerations

Christos G Massouros 1 and Naveed Yaqoob 2

Citation Massouros CG Yaqoob

N On the Theory of LeftRight

Almost Groups and Hypergroups

with their Relevant Enumerations



Academic Editor Askar Tuganbaev

Received 14 June 2021

Accepted 6 July 2021

Published 3 August 2021




iations








40)


2 Department of Mathematics and Statistics Riphah International University Sector I-14Islamabad 44000 Pakistan nayaqoobymailcom


Abstract This paper presents the study of algebraic structures equipped with the inverted asso-ciativity axiom Initially the definition of the left and the right almost-groups is introduced andafterwards the study is focused on the more general structures which are the left and the rightalmost-hypergroups and on their enumeration in the cases of order 2 and 3 The outcomes of theseenumerations compared with the corresponding in the hypergroups reveal interesting results Nextfundamental properties of the left and right almost-hypergroups are proved Subsequently thealmost hypergroups are enriched with more axioms like the transposition axiom and the weakcommutativity This creates new hypercompositional structures such as the transposition leftrightalmost-hypergroups the leftright almost commutative hypergroups the join leftright almosthypergroups etc The algebraic properties of these new structures are analyzed and studied as wellEspecially the existence of neutral elements leads to the separation of their elements into attractiveand non-attractive ones If the existence of the neutral element is accompanied with the existence ofsymmetric elements as well then the fortified transposition leftright almost-hypergroups and thetransposition polysymmetrical leftright almost-hypergroups come into being

Keywords hypercompositional algebra magma leftright almost-group leftright almost-hypergroup transposition axiom Mathematica

1 Introduction

This paper is generally classified in the area of hypercompositional algebra Hyper-compositional algebra is the branch of abstract algebra which studies hypercompositionalstructures ie structures equipped with one or more multi-valued operations Multi-valued operations also called hyperoperations or hypercompositions are operations inwhich the result of the synthesis of two elements is multi-valued rather than a singleelement More precisely a hypercomposition on a non-void set H is a function from H times Hto the powerset P(H) of H Hypercompositional structures came into being through thenotion of the hypergroup The hypergroup was introduced in 1934 by Marty in order tostudy problems in non-commutative algebra such as cosets determined by non-invariantsubgroups [1ndash3]

In [4] the magma is defined as an ordered pair (Hperp) where H is a non-void set andldquoperprdquo is a law of synthesis which is either a composition or a hypercomposition Per thisdefinition if A and B are subsets of H then

AperpB = aperpb isin H | a isin A b isin B if perp is a composition

andAperpB = cup

(ab)isinAtimesB(aperpb) if perp is a hypercomposition

191


In particular if A = empty or B = empty then AperpB = empty and vice versa Aperpb and aperpB havethe same meaning as Aperpb and aperpB In general the singleton a is identified withits member a Sometimes it is convenient to use the relational notation A asymp B instead ofA cap B 6= empty Then as the singleton a is identified with its member a the notation a asymp Aor A asymp a is used as a substitute for a isin A or Aa The relation asymp may be considered as aweak generalization of equality since if A and B are singletons and A asymp B then A = BThus a asymp bperpc means either a = bperpc when the synthesis is a composition or a isin bperpcwhen the synthesis is a hypercomposition

It is possible that the result of the synthesis of a pair of elements in a magma is thevoid set when the law of synthesis is a hypercomposition Then the structure is called apartial hypergroupoid otherwise it is called a hypergroupoid


ab = x isin E | a asymp xb

andba = x isin E | a asymp bx

Thus x asymp ab if and only if a asymp xb and x asymp ba if and only if a asymp bx In the case of amultiplicative magma the two induced laws are named inverse laws and they are called theright division and the left division respectively [4] It is obvious that if the law of synthesisis commutative then the right division and left division coincide

A law of synthesis on a set H is called associative if the property


is valid for all elements x y z in H A magma whose law of synthesis is associative iscalled an associative magma [4]

Definition 1 An associative magma in which the law of synthesis is a composition is called asemigroup while it is called a semihypergroup if the law of synthesis is a hypercomposition andab 6= empty for each pair of its elements

A law of synthesis (x y)rarr xperpy on a set H is called reproductive if the equality

xperpH = Hperpx = H

is valid for all elements x in H A magma whose law of synthesis is reproductive is called areproductive magma [4]

Definition 2 A reproductive magma in which the law of synthesis is a composition is called aquasigroup while it is called a quasi-hypergroup if the law of synthesis is a hypercomposition andab 6= empty for each pair of its elements

Definition 3 [4] An associative and reproductive magma is called a group if the law of synthesisis a composition while it is called a hypergroup if the law of synthesis is a hypercomposition

The above unified definition of the group and the hypergroup is presented in [4]where it is also proved analytically that it is equivalent to the well-known dominantdefinition of the group

A composition or a hypercomposition on a non-void set H is called left invertedassociative if

(aperpb)perpc = (cperpb)perpa for every a b c isin H

192


while it is called right inverted associative if

aperp(bperpc) = cperp(bperpa) for every a b c isin H

The notion of the inverted associativity was initially conceived by Kazim and Naseerud-din [5] who endowed a groupoid with the left inverted associativity thus defining theLA-semigroup A magma equipped with left inverted assisiativity is called a left invertedassociative magma while if it is equipped with right inverted assisiativity is called a rightinverted associative magma

Recall that if (Eperp) is a magma then the law of synthesis


is called the opposite of ldquoperprdquo The magma (Eperpop) is called the opposite magma of (Eperp) [4]

Theorem 1 If (Hperp) is a left inverted associative magma then (Hperpop) is a right invertedassociative magma

Proofaperpop(bperpopc) = (bperpopc)perpa = (cperpb)perpa =

= (aperpb)perpc = cperpop(aperpb) = cperpop(bperpopa)

As it is detailed in [4] the group and the hypergroup satisfy exactly the same axiomsie the reproductive axiom and the associative axiom This led to the unified definition of thegroup and the hypergroup that was repeated as Definition 3 in this paper Using the sameapproach the definition of the leftright almost-group and the leftright almost-hypergroup is

Definition 4 (FIRST DEFINITION OF THE LEFTRIGHT ALMOST GROUPHYPERGROUP) A reproductive magma which satisfies the axiom of the left inverted associativity is calleda left almost-group (LA-group) if the law of synthesis on the magma is a composition while it iscalled a left almost-hypergroup (LA-hypergroup) if the law of synthesis is a hypercomposition Areproductive right inverted associative magma is called a right almost-group (RA-group) or a rightalmost-hypergroup (RA-hypergroup) if the law of synthesis is a composition or a hypercomposi-tion respectively

Remark 1 Obviously if the law of synthesis is commutative then the LA- or RA- groups andhypergroups are groups and hypergroups respectively indeed

(aperpb)perpc = (cperpb)perpa = aperp(cperpb) = aperp(bperpc)

As shown in Theorem 11 in [4] when the law of synthesis is a composition then thereproductive axiom is valid if and only if the inverse laws are compositions Hence anotherdefinition can be given for the leftright almost-group

Definition 5 (SECOND DEFINITION OF THE LEFTRIGHT ALMOST-GROUP) A magmawhich satisfies the axiom of the leftright inverted associativity is called a leftright almost-group(LA-groupRA-group) if the law of synthesis on the magma is a composition and the two inducedlaws of synthesis are compositions as well

Thus if the law of synthesis on a magma is written multiplicatively then the magmais a leftright almost-group if and only if it satisfies the axiom of the leftright invertedassociativity and both the right and the left division ab ba respectively result in a singleelement for every pair of elements a b in the magma

193


In a similar way because of Theorem 14 in [4] if ab 6= empty and ba 6= empty for all pairs ofelements a b of a magma then the magma is reproductive Therefore a second definitionof the leftright almost-hypergroup can be given

Definition 6 (SECOND DEFINITION OF THE LEFTRIGHT ALMOST-HYPERGROUP)A magma which satisfies the axiom of the leftright inverted associativity is called a leftrightalmost-hypergroup (LA-hypergroupRA-hypergroup) if the law of synthesis on the magma is ahypercomposition and the result of each one of the two inverse hypercompositions is nonvoid for allpairs of elements of the magma

Example 1 This example proves the existence of non-trivial left almost-groups (Table 1) and rightalmost-groups (Table 2)

Table 1 Left almost-group


groupRA-group) if the law of synthesis on the magma is a composition and the two induced laws of synthesis are compositions as well

Thus if the law of synthesis on a magma is written multiplicatively then the magma is a leftright almost-group if and only if it satisfies the axiom of the leftright inverted associativity and both the right and the left division a b b a respectively result in a single element for every pair of elements a b in the magma

In a similar way because of Theorem 14 in [4] if a b ne empty and b a ne empty for all pairs of elements a b of a magma then the magma is reproductive Therefore a second definition of the leftright almost-hypergroup can be given

Definition 6 (SECOND DEFINITION OF THE LEFTRIGHT ALMOST-HYPERGROUP) A magma which satisfies the axiom of the leftright inverted associativity is called a leftright almost-hypergroup (LA-hypergroupRA-hypergroup) if the law of synthesis on the magma is a hypercomposition and the result of each one of the two inverse hypercompositions is nonvoid for all pairs of elements of the magma

Example 1 This example proves the existence of non-trivial left almost-groups (Table 1) and right almost-groups (Table 2)


1 2 3

1 1 2 3

2 3 1 2

3 2 3 1

Table 2 Right almost-group

1 2 3

1 1 3 2

2 2 1 3

3 3 2 1









1 2 3

1 1 2 3

2 3 1 2

3 2 3 1


1 2 3

1 1 3 2

2 2 1 3

3 3 2 1

Example 2 This example presents a non-trivial left almost-hypergroup (Table 3) and a non-trivialright almost-hypergroup (Table 4)

Table 3 Left almost-hypergroup


Example 2 This example presents a non-trivial left almost-hypergroup (Table 3) and a non-trivial right almost-hypergroup (Table 4)


1 2 3

1 1 1 123

2 1 1 23

3 23 23 123

Table 4 Right almost-hypergroup

1 2 3

1 2 3 13

2 12 13 123

3 13 123 123

Remark 2 It is noted that in the groups the unified definition which uses only the reproductive axiom and the associative axiom leads to the existence of a bilaterally neutral element and consequently to the existence of a symmetric element for each one of the grouprsquos elements as it is proved in Theorem 2 in [4] Nothing analogous holds in the case of the left or right almost-groups Indeed if e is a neutral element in a left almost-group then

( )a e c a cperp perp = perp and ( ) ( )a e c c e a c aperp perp = perp perp = perp

Hence the composition is commutative and therefore the left almost-group is a group

A direct consequence of Theorem 1 is the following theorem

Theorem 2 If ( )H perp is a left almost-group or hypergroup then ( ) opH perp is a right almost-

group or hypergroup respectively

Corollary 1 The transpose of the Cayley table of a left almost-group or hypergroup is the Cayley table of a right almost-group or hypergroup respectively and vice versa

Example 3 The transpose of the Cayley Table 5 which describes a LA-hypergroup is Table 6 which describes a RA-hypergroup

194






1 2 3

1 1 1 123

2 1 1 23

3 23 23 123


1 2 3

1 2 3 13

2 12 13 123

3 13 123 123









Remark 2 It is noted that in the groups the unified definition which uses only the reproductiveaxiom and the associative axiom leads to the existence of a bilaterally neutral element and conse-quently to the existence of a symmetric element for each one of the grouprsquos elements as it is provedin Theorem 2 in [4] The same doesnrsquot hold in the case of the left or right almost-groups Indeed if eis a neutral element in a left almost-group then

(aperpe)perpc = aperpc and (aperpe)perpc = (cperpe)perpa = cperpa



Theorem 2 If (Hperp) is a left almost-group or hypergroup then (Hperpop) is a right almost-groupor hypergroup respectively

Corollary 1 The transpose of the Cayley table of a left almost-group or hypergroup is the Cayleytable of a right almost-group or hypergroup respectively and vice versa

Example 3 The transpose of the Cayley Table 5 which describes a LA-hypergroup is Table 6 whichdescribes a RA-hypergroup




1 2 3

1 1 12 13

2 13 123 3

3 12 2 123


1 2 3

1 1 13 12

2 12 123 2

3 13 3 123

Corollary 2 The cardinal number of the left almost-groups which are defined over a set H is equal to the cardinal number of the right almost-groups defined over H

Corollary 3 The cardinal number of the left almost-hypergroups which are defined over a set H is equal to the cardinal number of the right almost-hypergroups defined over H

Hypercompositional structures with inverted associativity were studied by many authors (eg Yaqoob et al [6ndash10] Hila et al [1112] etc)

Special notation In the following in addition to the typical algebraic notations we use Krasner rsquos notation for the complement and difference So we denote with AmiddotmiddotB the set of elements that are in the set A but not in the set B

2 Hypercompositional Structures with Inverted Associativity The hypergroup was progressively enriched with additional axioms thus creating a

significant number of special hypergroups eg [13ndash55] Prenowitz enriched the hypergroup with an additional axiom in order to use it in the study of geometry [56ndash61] More precisely he introduced the transposition axiom

a b c dcap ne empty implies ad bccap ne empty for all a b c d Hisin

into a commutative hypergroup of idempotent elements ie a commutative hypergroup H all the elements of which satisfy the properties aa a= and a a a= He named this new hypergroup join space Prenowitz was followed by others such as Jantosciak [6263] Barlotti and Strambach [64] Freni [6566] Massouros [67ndash71] Dramalidis [72] etc For the sake of terminology unification the commutative hypergroups which satisfy the transposition axiom are called join hypergroups It has been proved that the join hypergroups also comprise a useful tool in the study of languages and automata [35ndash387374] Later on Jantosciak generalized the above axiom in an arbitrary hypergroup as follows

b a c dcap ne empty implies ad bccap ne empty for all a b c d Hisin

195





1 2 3

1 1 12 13

2 13 123 3

3 12 2 123


1 2 3

1 1 13 12

2 12 123 2

3 13 3 123










Corollary 2 The cardinal number of the left almost-groups which are defined over a set H is equalto the cardinal number of the right almost-groups defined over H

Corollary 3 The cardinal number of the left almost-hypergroups which are defined over a set H isequal to the cardinal number of the right almost-hypergroups defined over H

Hypercompositional structures with inverted associativity were studied by manyauthors (eg Yaqoob et al [6ndash10] Hila et al [1112] etc)

Special notation In the following in addition to the typical algebraic notations we useKrasnerrsquos notation for the complement and difference So we denote with AmiddotmiddotB the set ofelements that are in the set A but not in the set B

2 Hypercompositional Structures with Inverted Associativity

The hypergroup was progressively enriched with additional axioms thus creatinga significant number of special hypergroups eg [13ndash55] Prenowitz enriched the hyper-group with an additional axiom in order to use it in the study of geometry [56ndash61] Moreprecisely he introduced the transposition axiom

ab cap cd 6= empty implies ad cap bc 6= empty for all a b c d isin H

into a commutative hypergroup of idempotent elements ie a commutative hypergroupH all the elements of which satisfy the properties aa = a and aa = a He named this newhypergroup join space Prenowitz was followed by others such as Jantosciak [6263] Barlottiand Strambach [64] Freni [6566] Massouros [67ndash71] Dramalidis [72] etc For the sakeof terminology unification the commutative hypergroups which satisfy the transpositionaxiom are called join hypergroups It has been proved that the join hypergroups also comprisea useful tool in the study of languages and automata [35ndash387374] Later on Jantosciakgeneralized the above axiom in an arbitrary hypergroup as follows


He named this particular hypergroup transposition hypergroup [33]A quasicanonical hypergroup or polygroup [26ndash29] is a transposition hypergroup H

containing a scalar identity that is there exists an element e such that ea = ae = a for each ain H A canonical hypergroup [17ndash19] is a commutative polygroup A canonical hypergroupmay also be characterized as a join hypergroup with a scalar identity [23] The followingProposition connects the canonical hypergroups with the RA-hypergroups

Proposition 1 Let (H middot) be a canonical hypergroup and ldquordquo the induced hypercomposition whichfollows from ldquomiddotrdquo Then (H ) is a right almost-hypergroup

196


Proof In a canonical hypergroup (H middot) each element x isin H has a unique inverse which isdenoted by xminus1 Moreover ex = xminus1 and yx = yxminus1 [43234] Thus

a(bc)= a(

bcminus1)= a

(bcminus1

)minus1= a

(cbminus1

)= (ac)bminus1 = (ca)bminus1 = c

(abminus1

)=

= c(

baminus1)minus1

= cbaminus1 = c(ba)

Hence the right inverted associativity is valid Moreover in any hypergroup holds [475]

aH = Ha = H

Thus the reproductive axiom is valid and therefore (H ) is a RA-hypergroup

Subsequently the left and right almost-hypergroups can be enriched with additionalaxioms The first axiom to be used for this purpose is the transposition axiom as it hasbeen introduced into many hypercompositional structures and has given very interestingand useful properties (eg see [7576])

Definition 7 If a left almost-hypergroup (H middot) satisfies the transposition axiom ie

ba cap cd 6= empty implies ad cap bc 6= empty f or all a b c d isin H

then it will be called transposition left almost-hypergroup If a right almost-hypergroup satisfies thetransposition axiom then it will be called transposition right almost-hypergroup

Example 4 This example presents a transposition left almost-hypergroup (Table 7) and a transpo-sition right almost-hypergroup (Table 8)

Table 7 Transposition left almost-hypergroup


He named this particular hypergroup transposition hypergroup [33] A quasicanonical hypergroup or polygroup [26ndash29] is a transposition hypergroup H

containing a scalar identity that is there exists an element e such that = =ea ae a for each a in H A canonical hypergroup [17ndash19] is a commutative polygroup A canonical hypergroup may also be characterized as a join hypergroup with a scalar identity [23] The following Proposition connects the canonical hypergroups with the RA-hypergroups

Proposition 1 Let ( )H sdot be a canonical hypergroup and ldquordquo the induced hypercomposition

which follows from ldquo sdot rdquo Then ( ) H is a right almost-hypergroup

Proof In a canonical hypergroup ( )H sdot each element x Hisin has a unique inverse

which is denoted by 1xminus Moreover 1e x xminus= and 1y x yxminus= [43234] Thus

( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )

11 1 1 1 1 1

11 1

a b c a bc a bc a cb ac b ca b c ab

c ba c ba c b a

minusminus minus minus minus minus minus

minusminus minus

= = = = = = =

= = =

Hence the right inverted associativity is valued Moreover in any hypergroup holds [475]

a H H a H= =

Thus the reproductive axiom is valid and therefore ( ) H is a RA-hypergroup

Subsequently the left and right almost-hypergroups can be enriched with additional axioms The first axiom to be used for this purpose is the transposition axiom as it has been introduced into many hypercompositional structures and has given very interesting and useful properties (eg see [7576])

Definition 7 If a left almost-hypergroup ( )H sdot satisfies the transposition axiom ie

cap ne empty b a c d implies cap ne emptyad bc for all isin a b c d H

then it will be called transposition left almost-hypergroup If a right almost-hypergroup satisfies the transposition axiom then it will be called transposition right almost-hypergroup

Example 4 This example presents a transposition left almost-hypergroup (Table 7) and a transposition right almost-hypergroup (Table 8)


1 2 3

1 123 123 123

2 13 12 13

3 23 23 1

Table 8 Transposition right almost-hypergroup Table 8 Transposition right almost-hypergroup


1 2 3

1 12 123 13

2 12 13 123

3 123 12 13

In [4] the reverse transposition axiom was introduced

cap ne emptyad bc implies cap ne empty b a c d for all isin a b c d H

Thus the following hypercompositional structure can be defined

Definition 8 If a left (right) almost-hypergroup ( )H sdot satisfies the reverse transposition axiom then it will be called a reverse transposition left (right) almost-hypergroup

Definition 9 A hypercomposition on a non-void set H is called left inverted weakly associative if

( ) ( )ab c cb acap ne empty for every a b c Hisin

while it is called right inverted weakly associative if

( ) ( )a bc c bacap ne empty for every a b c Hisin

Definition 10 A quasi-hypergroup equipped with a hypercomposition which is left inverted weakly associative is called a weak left almost-hypergroup (WLA-hypergroup) while it is called a weak right almost-hypergroup (WRA-hypergroup) if the hypercomposition is right inverted weakly associative

Recall that a quasi-hypergroup which satisfies the weak associativity is called HV-group [77]

Proposition 2 A commutative WLA-hypergroup (or WRA-hypergroup) is a commutative HV-group

Proof Suppose that H is a commutative WLA-hypergroup then

( ) ( ) ( ) ( ) ( ) ( ) ab c cb a ab c a cb ab c a bccap ne empty hArr cap ne empty hArr cap ne empty

Hence H is an HV-group

Proposition 3 Let ( )H sdot be a left almost-hypergroup An arbitrary subset Iab of H is associated

to each pair of elements ( ) 2a b Hisin and the following hypercomposition is introduced into H

aba b ab Ilowast = cup

Then ( )H lowast is a WLA-hypergroup

Proof Since xH and Hx are subsets of xlowastH and Hlowastx respectively it follows that the reproductive axiom holds On the other hand




197


Definition 8 If a left (right) almost-hypergroup (H middot) satisfies the reverse transposition axiomthen it will be called a reverse transposition left (right) almost-hypergroup


(ab)c cap (cb)a 6= empty for every a b c isin H


a(bc) cap c(ba) 6= empty for every a b c isin H

Definition 10 A quasi-hypergroup equipped with a hypercomposition which is left inverted weaklyassociative is called a weak left almost-hypergroup (WLA-hypergroup) while it is called a weak rightalmost-hypergroup (WRA-hypergroup) if the hypercomposition is right inverted weakly associative




(ab)c cap (cb)a 6= emptyhArr (ab)c cap a(cb) 6= emptyhArr (ab)c cap a(bc) 6= empty


Proposition 3 Let (H middot) be a left almost-hypergroup An arbitrary subset Iab of H is associated toeach pair of elements (a b) isin H2 and the following hypercomposition is introduced into H

a lowast b = ab cup Iab

Then (H lowast) is a WLA-hypergroup

Proof Since xH and Hx are subsets of xlowastH and Hlowastx respectively it follows that thereproductive axiom holds On the other hand

(a lowast b) lowast c= (ab cup Iab) lowast c = (ab) lowast c cup Iab lowast c =[cup

risinab(rc cup Irc)

]cup[cup

sisinIab(sc cup Isc)

]=

= (ab)c cup[cup

risinabIrc

]cup[cup


]

and

(c lowast b) lowast a= (cb cup Icb) lowast a = (cb) lowast a cup Icb lowast a =

[cup

risincb(ra cup Ira)

]cup[cup

sisinIcb(sa cup Isa)

]=

= (cb)a cup[cup

risincbIra

]cup[cup


]

Since (ab)c = (cb)a it follows that a lowast (b lowast c) cap (a lowast b) lowast c 6= empty

Proposition 4 If capabisinH

Iab 6= empty then (H lowast) is a transposition WLA-hypergroup

It is obvious that if the composition or the hypercomposition is commutative then theinverted associativity coincides with the associativity Thus a commutative LA-hypergroupis simply a commutative hypergroup However in the hypercompositions there exists a

198


property that does not appear in the compositions This is the weak commutativity Ahypercomposition on a non-void set H is called weakly commutative if

ab cap ba 6= empty for all a b isin H

Definition 11 A left almost commutative hypergroup (LAC-hypergroup) is a left almost-hypergroupwhich satisfies the weak commutativity A right almost commutative hypergroup (RAC-hypergroup)is a right almost-hypergroup which satisfies the weak commutativity A LAC-hypergroup (respRAC-hypergroup) which satisfies the transposition axiom will be called join left almost-hypergroup(resp reverse join left almost-hypergroup) A LAC-hypergroup (resp RAC-hypergroup) whichsatisfies the reverse transposition axiom will be called reverse join left almost-hypergroup (respreverse join right almost-hypergroup)

Example 5 This example presents a join left almost-hypergroup (Table 9) and a join right almost-hypergroup (Table 10)

Table 9 Join left almost-hypergroup


( ) ( ) ( ) ( ) ( )

( ) ( )

ab

ab

ab ab rc scr ab s I

rc scr ab s I

a b c ab I c ab c I c rc I sc I

ab c I sc I

isin isin

isin isin

= cup = cup = cup cup =

= cup

and

( ) ( ) ( ) ( ) ( )

( ) ( )

cb

cb

cb cb ra sar cb s I

ra sar cb s I

c b a cb I a cb a I a ra I sa I

cb a I sa I

isin isin

isin isin


= cup

Since ( ) ( )ab c cb a= it follows that ( ) ( ) a b c a b ccap ne empty

Proposition 4 If

aba b H

Iisin

ne empty then (H lowast) is a transposition WLA-hypergroup

It is obvious that if the composition or the hypercomposition is commutative then the inverted associativity coincides with the associativity Thus a commutative LA-hypergroup is simply a commutative hypergroup However in hypercompositions there exists a property that does not appear in the compositions This is the weak commutativity A hypercomposition on a non-void set H is called weakly commutative if

ab bacap ne empty for all a b Hisin

Definition 11 A left almost commutative hypergroup (LAC-hypergroup) is a left almost-hypergroup which satisfies the weak commutativity A right almost commutative hypergroup (RAC-hypergroup) is a right almost-hypergroup which satisfies the weak commutativity A LAC-hypergroup (resp RAC-hypergroup) which satisfies the transposition axiom will be called join left almost-hypergroup (resp reverse join left almost-hypergroup) A LAC-hypergroup (resp RAC-hypergroup) which satisfies the reverse transposition axiom will be called reverse join left almost-hypergroup (resp reverse join right almost-hypergroup)



1 2 3

1 23 123 13

2 12 13 123

3 13 23 12

Table 10 Join right almost-hypergroup Table 10 Join right almost-hypergroup


1 2 3

1 13 12 123

2 123 13 12

3 13 12 23

A weak left almost-hypergroup which satisfies the weak commutativity will be named a weak left almost commutative hypergroup (WLAC-hypergroup) Analogous is the definition of the weak right almost commutative hypergroup (WRAC-hypergroup)

Proposition 5 If ( )H sdot is a LAC-hypergroup then ( )H lowast is WLAC-hypergroup

Proposition 6 Let ( )H sdot be a left almost-hypergroup and ab baI I= for all a b Hisin then

( )H lowast is a WLAC-hypergroup

Proposition 7

i If ( )H sdot is a LAC-hypergroup and

aba b H

Iisin

ne empty then ( )H lowast is a weak join left

almost-hypergroup ii If ( )H sdot is a LA-hypergroup ab baI I= for all a b Hisin and

ab

a b HI

isinne empty then

( )H lowast is a weak join left almost-hypergroup

Corollary 4 If H is a left almost -hypergroup and w is an arbitrary element of H then H endowed with the hypercomposition

x y xy x y wlowast = cup

is a weak join left almost-hypergroup

Remark 3 Analogous propositions to the above 3ndash7 hold for the right almost-hypergroups as well

3 Enumeration and Structure results The enumeration of hypercompositional structures is the subject of several papers

(eg [78ndash87]) In [78] a symbolic manipulation package is developed which enumerates the hypergroups of order 3 separates them into isomorphism classes and calculates their cardinality Following analogous techniques in this paper a package is developed (see Appendix A) which when combined with the package in [78] enumerates the left almost-hypergroups and the right almost-hypergroups with 3 elements classifies them in isomorphism classes and computes their cardinality

For the purpose of the package the set 123H = is used as the set with three elements The laws of synthesis in H are defined through the Cayley Table 11rdquo

A weak left almost-hypergroup which satisfies the weak commutativity will be nameda weak left almost commutative hypergroup (WLAC-hypergroup) Analogous is the definitionof the weak right almost commutative hypergroup (WRAC-hypergroup)

Proposition 5 If (H middot) is a LAC-hypergroup then (H lowast) is WLAC-hypergroup

Proposition 6 Let (H middot) be a left almost-hypergroup and Iab = Iba for all a b isin H then (H lowast)is a WLAC-hypergroup

Proposition 7

i If (H middot) is a LAC-hypergroup and capabisinH

Iab 6= empty then (H lowast) is a weak join left almost-

hypergroupii If (H middot) is a LA-hypergroup Iab = Iba for all a b isin H and cap

abisinHIab 6= empty then (H lowast) is a

weak join left almost-hypergroup

199


Corollary 4 If H is a left almost-hypergroup and w is an arbitrary element of H then H endowedwith the hypercomposition

x lowast y = xy cup x y wis a weak join left almost-hypergroup


3 Enumeration and Structure Results

The enumeration of hypercompositional structures is the subject of several papers(eg [78ndash87]) In [78] a symbolic manipulation package is developed which enumer-ates the hypergroups of order 3 separates them into isomorphism classes and calculatestheir cardinality Following analogous techniques in this paper a package is developed(see Appendix A) which when combined with the package in [78] enumerates the leftalmost-hypergroups and the right almost-hypergroups with 3 elements classifies them inisomorphism classes and computes their cardinality

For the purpose of the package the set H = 1 2 3 is used as the set with threeelements The laws of synthesis in H are defined through the Cayley Table 11

Table 11 General form of a 3-element magmarsquos Cayley table



1 2 3

1 a11 a12 a13

2 a21 a22 a23

3 a31 a32 a33

where the intersection of row i with column j ie aij is the result of ij Αs in the case of hypergroups in the left almost-hypergroups or right almost-hypergroups the result of the hypercomposition of any two elements is non-void (see Theorem 4 below) Thus the cardinality of the set of all possible magmas with 3 elements which are not partial hypergroupoids is 79 = 40 353 607 As it is mentioned above in the commutative case the left inverted associativity and the right inverted associativity coincide with the associativity Among the 40 353 607 magmas only 2 520 are commutative hypergroups which are the trivial cases of left almost-hypergroups and right almost-hypergroups Thus the package focuses on the non-trivial cases that is on the non-commutative magmas The enumeration reveals that there exist 65 955 reproductive non-commutative magmas which satisfy the left inverted associativity only and obviously the same number of reproductive non-commutative magmas which satisfy the right inverted associativity only That is there exist 65 955 non-trivial left almost-hypergroups and 65 955 non-trivial right almost-hypergroups Moreover there exist 7 036 reproductive magmas which satisfy both the left and the right inverted associativity ie there exist 7 036 both left and right almost-hypergroups Furthermore there are 16 044 reproductive magmas that satisfy the left inverted associativity the right inverted associativity and the associativity This means that there exist 16 044 structures which are simultaneously left almost-hypergroups right almost-hypergroups and hypergroups Finally there do not exist any reproductive magmas which satisfies both the left (or right) inverted associativity and the associativity

The following examples present the worth mentioning cases in which a hypercompositional structure is (a) both left and right almost-hypergroup and (b) simultaneously left almost-hypergroup right almost-hypergroup and non-commutative hypergroup

Example 6 The hypercompositional structure described in Cayley Table 12 is both left and right almost-hypergroup

Table 12 A LA- and RA-hypergroup

1 2 3

1 1 12 123

2 12 3 13

3 123 123 123

Example 7 The hypercompositional structure described in Cayley Table 13 is simultaneously left almost-hypergroup right almost-hypergroup and non-commutative hypergroup

where the intersection of row i with column j ie aij is the result of ij As in the caseof hypergroups in the left almost-hypergroups or right almost-hypergroups the resultof the hypercomposition of any two elements is non-void (see Theorem 4 below) Thusthe cardinality of the set of all possible magmas with 3 elements which are not partialhypergroupoids is 79 = 40 353 607 As it is mentioned above in the commutative case theleft inverted associativity and the right inverted associativity coincide with the associativityAmong the 40 353 607 magmas only 2520 are commutative hypergroups which are thetrivial cases of left almost-hypergroups and right almost-hypergroups Thus the packagefocuses on the non-trivial cases that is on the non-commutative magmas The enumera-tion reveals that there exist 65 955 reproductive non-commutative magmas which satisfythe left inverted associativity only and obviously the same number of reproductive non-commutative magmas which satisfy the right inverted associativity only That is there exist65 955 non-trivial left almost-hypergroups and 65 955 non-trivial right almost-hypergroupsMoreover there exist 7036 reproductive magmas which satisfy both the left and the rightinverted associativity ie there exist 7036 both left and right almost-hypergroups Fur-thermore there are 16 044 reproductive magmas that satisfy the left inverted associativitythe right inverted associativity and the associativity This means that there exist 16 044structures which are simultaneously left almost-hypergroups right almost-hypergroupsand hypergroups Finally there do not exist any reproductive magmas which satisfies boththe left (or right) inverted associativity and the associativity

The following examples present the worth mentioning cases in which a hypercompo-sitional structure is (a) both left and right almost-hypergroup and (b) simultaneously leftalmost-hypergroup right almost-hypergroup and non-commutative hypergroup

200


Example 6 The hypercompositional structure described in Cayley Table 12 is both left and rightalmost-hypergroup




1 2 3

1 a11 a12 a13

2 a21 a22 a23

3 a31 a32 a33





1 2 3

1 1 12 123

2 12 3 13

3 123 123 123

Example 7 The hypercompositional structure described in Cayley Table 13 is simultaneously left almost-hypergroup right almost-hypergroup and non-commutative hypergroup Example 7 The hypercompositional structure described in Cayley Table 13 is simultaneously leftalmost-hypergroup right almost-hypergroup and non-commutative hypergroup

Table 13 A non-commutative hypergroup which is also left and right almost-hypergroup



1 2 3

1 1 1 123

2 1 1 23

3 123 123 123

A magma though with three elements can be isomorphic to another magma which results from a permutation of these three elements The isomorphic structures which appear in this way can be considered as members of the same class These classes can be constructed and enumerated with the use of the techniques and methods which are developed in [78] Having done so the following conclusions occurred

bull The 65 955 non-trivial left almost-hypergroups are partitioned in 11 067 isomorphism classes 10 920 of them consist of 6 members 142 have 3 members 4 have 2 members and the last one is a one-member class The same are valid for the 65 955 non-trivial right almost-hypergroups

bull The 7 036 both left and right almost-hypergroups are partitioned in 1 174 isomorphism classes 1 172 of them consist of 6 members while the other 2 have 2 members

bull The 16 044 noncommutative structures which are simultaneously left almost-hypergroups right almost-hypergroups and hypergroups are partitioned in 2 733 isomorphism classes 2 617 of them consist of 6 members 110 have 3 members and the last 6 have 2 members

A magma though with three elements can be isomorphic to another magma whichresults from a permutation of these three elements The isomorphic structures whichappear in this way can be considered as members of the same class These classes canbe constructed and enumerated with the use of the techniques and methods which aredeveloped in [78] Having done so the following conclusions occurred

bull The 65 955 non-trivial left almost-hypergroups are partitioned in 11 067 isomorphismclasses 10 920 of them consist of 6 members 142 have 3 members 4 have 2 membersand the last one is a one-member class The same are valid for the 65 955 non-trivialright almost-hypergroups

bull The 7036 both left and right almost-hypergroups are partitioned in 1174 isomorphismclasses 1172 of them consist of 6 members while the other 2 have 2 members

bull The 16 044 noncommutative structures which are simultaneously left almost-hypergroups right almost-hypergroups and hypergroups are partitioned in 2733 isomor-phism classes 2617 of them consist of 6 members 110 have 3 members and the last 6have 2 members

The above results combined with the ones of [78] for the hypergroups are summa-rized and presented in the following Table 14

201


Table 14 Classification of the LA-hypergroups and the RA-hypergroups with three elements

TotalNumber

IsomorphismClasses

Classes with1 Member

(Rigid)

Classeswith 2

Members

Classeswith 3

Members

Classes with4 or 5

Members

Classeswith 6

Membersnon-trivial left

almost-hypergroups 65 955 11 067 1 4 142 0 10 920

non-trivial rightalmost-hypergroups 65 955 11 067 1 4 142 0 10 920

non-trivial both leftand right

almost-hypergroups7036 1174 0 2 0 0 1172

simultaneously leftalmost-hypergroups

rightalmost-hypergroups

and non-commutativehypergroups

16 044 2733 0 6 110 0 2617

non-commutativehypergroups

(satisfying theassociativity only)

4628 800 0 1 56 0 723



0

simultaneously rightalmost-hypergroups


0

commutativehypergroups (trivial

leftalmost-hypergroups

and trivial rightalmost-hypergroups)

2520 466 6 3 78 0 399

hypergroups 23 192 3999 6 10 244 0 3739

In the above table we observe that there is only one class of left almost-hypergroupsand only one class of right almost-hypergroups which contains a single member Themember of this class is of particular interest since its automorphism group is of order 1Such hypercompositional structures are called rigid Additionally observe that there are6 rigid hypergroups all of which are commutative A study and enumeration of theserigid hypergroups as well as other rigid hypercompositional structures is given in [81]The following Table 15 presents the unique rigid left almost-hypergroup while Table 16describes the unique rigid right almost-hypergroup with 3 elements

Table 15 The rigid left almost-hypergroup of three elements


hypergroups 23 192 3 999 6 10 244 0 3 739

In the above table we observe that there is only one class of left almost-hypergroups and only one class of right almost-hypergroups which contains a single member The member of this class is of particular interest since its automorphism group is of order 1 Such hypercompositional structures are called rigid Additionally observe that there are 6 rigid hypergroups all of which are commutative Α study and enumeration of these rigid hypergroups as well as other rigid hypercompositional structures is given in [81] The following Table 15 presents the unique rigid left almost-hypergroup while Table 16 describes the unique rigid right almost-hypergroup with 3 elements


1 2 3

1 123 23 23

2 13 123 13

3 12 12 123

Table 16 The rigid right almost-hypergroup of three elements

1 2 3

1 123 13 12

2 23 123 12

3 23 13 123

More generally the next theorem is valid

Theorem 3 A non-void set H equipped with the hypercomposition

H if x y

xyH x if x y == sdot sdot ne

becomes a rigid left almost-hypergroup while equipped with the hypercomposition

H if x y

xyH y if x y == sdot sdot ne

becomes a rigid right almost-hypergroup Both of them satisfy the transposition axiom

Remark 4 There exist 81 hypergroupoids of two elements 3 of which are LA- hypergroups 3 are RA- hypergroups 4 are commutative hypergroups and 4 are non-commutative hypergroups The enumeration of the two-element crisp and fuzzy hypergroups can be found in [82]

202




hypergroups 23 192 3 999 6 10 244 0 3 739



1 2 3

1 123 23 23

2 13 123 13

3 12 12 123


1 2 3

1 123 13 12

2 23 123 12

3 23 13 123



H if x y



H if x y






xy =

H i f x = yH middot middotx i f x 6= y


xy =

H i f x = yH middot middoty i f x 6= y


Remark 4 There exist 81 hypergroupoids of two elements 3 of which are LA-hypergroups 3 areRA-hypergroups 4 are commutative hypergroups and 4 are non-commutative hypergroups Theenumeration of the two-element crisp and fuzzy hypergroups can be found in [82]

4 Algebraic Properties

In hypercompositional algebra it is dominant that a hypercomposition on a set E is amapping of Etimes E to the non-void subsets of E In [4] it is shown that this restriction isnot necessary since it can be proved that the result of the hypercomposition is non void inmany hypercompositional structures Such is the hypergroup (see [4] Theorem 12 and [75]Property 11) the weakly associative magma ([4] Proposition 5) and consequently theHV-group ([76] Proposition 31) The next theorem shows that in the case of the leftrightalmost hypergroups the result of the hypercomposition is non-void as well The proof issimilar to the one in [4] Theorem 12 but not the same due to the validity of the invertedassociativity in these structures instead of the associativity

Theorem 4 In a left almost-hypergroup or in a right almost-hypergroup H the result of thehypercomposition of any two elements is non-void

Proof Suppose that H is a left almost-hypergroup and cb = empty for some c b in H Becauseof the reproductivity the equalities H = Hc = cH and H = Hb = bH are valid Then theleft inverted associativity gives

H = Hc = (Hb)c = (cb)H = emptyH = empty

which is absurd Next assume that H is a right almost-hypergroup and bc = empty for someb c in H The right inverted associativity gives

H = cH = c(bH) = H(bc) = Hempty = empty

which is absurd

203


Proposition 8 A left almost-hypergroup H is a hypergroup if and only if a(bc) = (cb)a holds forall a b c in H

Proof Let H be a hypergroup Then the associativity (ab)c = a(bc) holds for all a b cin H Moreover because of the assumption a(bc) = (cb)a is valid for all a b c in HTherefore (ab)c = (cb)a thus H is a left almost-hypergroup Conversely now suppose thata(bc) = (cb)a holds for all a b c in H Since H is a left almost-hypergroup the sequence ofthe equalities a(bc) = (cb)a = (ab)c is valid Consequently H is a hypergroup

Theorem 5 In any left or right almost-hypergroup H the following are valid

i ab 6= empty and ba 6= empty for all a b in Hii H = Ha = aH and H = aH = Ha for all a in Hiii the non-empty result of the induced hypercompositions is equivalent to the reproductive axiom

Proof (i) Because of the reproduction Hb = H for all b isin H Hence for every a isin H thereexists x isin H such that a isin xb Thus x isin ab and therefore ab 6= empty Similarly ba 6= empty

(ii) Because of Theorem 4 the result of the hypercomposition in H is always a non-empty set Thus for every x isin H there exists y isin H such that y isin xa which implies thatx isin ya Hence H sube Ha Moreover Ha sube H Therefore H = Ha Next let x isin HSince H = xH there exists y isin H such that a isin xy which implies that x isin ay HenceH sube aH Moreover aH sube H Therefore H = aH

(iii) Suppose that xa 6= empty for all a x isin H Thus there exists y isin H such thatx isin ya Therefore x isin Ha for all x isin H and so H sube Ha Next since Ha sube H for alla isin H it follows that H = Ha Per duality H = aH Conversely now per Theorem 4 thereproductivity implies that ab 6= empty and ba 6= empty for all a b in H

Proposition 9 In any left or right almost-hypergroup H the following are valid

i a(bc) cup a(cb) sube (ab)c andii (cb)a cup (bc)a sube c(ba)for all a b c in H

Proof (i) Let x isin a(bc) Then bc cap ax 6= empty (1) Next if x isin a(cb) then a isin x(cb)or cb cap xa 6= empty (2) From both (1) and (2) it follows that xc cap ab 6= empty So there existsz isin ab such that z isin xc which implies that x isin zc Hence x isin (ab)c Thus (i) is validSimilar is the proof of (ii)

Corollary 5 If A B C are non-empty subsets of any left or right almost-hypergroup H then thefollowing are valid

i A(BC) cup A(CB) sube (AB)C andii (CB)A cup (BC)A sube C(BA)

Proposition 10 Let a b c d be arbitrary elements of any left or right almost-hypergroup H Thenthe following are valid

i (ba)(cd) sube (bac)d cap b(acd)ii (ba)(dc) sube (bac)diii (cd)(ab) sube d(acb)

Proof (i) Let x isin (ba)(cd) Then because of Proposition 9i there exists y isin ba suchthat x isin y(cd) sube (yc)d Thus xd cap yc 6= empty or xd cap (ba)c 6= empty Because of Proposition9ii it holds that (ba)c sube bac Therefore x isin (bac)d (1) Next since x isin (ba)(cd)there exists z isin cd such that x isin (ba)z Because of Proposition 9ii the inclusion relation(ba)z sube b(a z) holds Thus bx cap az 6= empty or equivalently bx cap a(cd) 6= empty or because ofProposition 9i bx cap acd 6= empty Therefore x isin b(acd) (2) Now (1) and (2) give (i)

204


(ii) Suppose that x isin (ba)(dc) Then there exists y isin ba such thatx isin y(dc) sube (yc)d Thus xd cap yc 6= empty or equivalently xd cap (ba)c 6= empty But accordingto Proposition 9ii the inclusion (ba)c sube bac holds Hence x isin (bac)d

(iii) can be proved in a similar manner

Corollary 6 If A B C D are non-empty subsets of any left or right almost-hypergroup H thenthe following are valid

i (BA)(CD) sube (BAC)D cap B(ACD)ii (BA)(DC) sube (BAC)Diii (CD)(AB) sube D(ACB)

Proposition 11 Let a b be elements of a left or right almost-hypergroup H then

i b isin (ab)a andii b isin a(ba)

Proof (i) Let x isin ab Then a isin xb Hence b isin xa Thus b isin (ab)a Therefore (i) isvalid The proof of (ii) is similar

Corollary 7 If A B are non-empty subsets of any left or right almost-hypergroup H then

i B sube (AB)A andii B sube A(BA)

Remark 5 The above properties are consequences of the reproductive axiom and therefore are validin both the left and the right almost-hypergroups as well as in the hypergroups [39]

Proposition 12

i In any left almost-hypergroup the following property is valid

(ab)c = (bc)a (mixed left inverted associativity)

ii In any right almost-hypergroup the following property is valid

c(ba) = a(cb) (mixed right inverted associativity)

Proof (i) Let x isin (ab)c Then the following sequence of equivalent statements is valid

x isin (ab)chArr xc cap ab 6= emptyhArr a isin (xc)bhArr a isin (bc)x hArr x isin (bc)a

Similar is the proof of (ii)

Corollary 8

i If A B C are non-empty subsets of a left almost-hypergroup H then

(AB)C = (BC)A

ii If A B C are non-empty subsets of a right almost-hypergroup H then

C(BA) = A(CB)

Proposition 13

i In any left almost-hypergroup the right inverted associativity of the induced hypercompositionsis valid

b(ac) = c(ab)

205


ii In any right almost-hypergroup the left inverted associativity of the induced hypercompositionsis valid

(ba)c = (ca)b

Proof For (i) it holds that

b(ac) = x isin H | bx cap ac 6= empty = x isin H | a isin (bx)c == x isin H | a isin (bx)c = x isin H | ab cap cx 6= empty = c(ab)

Regarding (ii) it is true that

(ba)c = x isin H | ba cap xc 6= empty = x isin H | a isin b(xc) == x isin H | a isin c(xb) = x isin H | ca cap xb 6= empty = (ca)b

This completes the proof

Corollary 9


B(AC) = C(AB)


(BA)C = (CA)B

5 Identities and Symmetric Elements

Let H be a non-void set endowed with a hypercomposition An element e of H iscalled right identity if x isin x middot e for all x in H If x isin e middot x for all x in H then e is calledleft identity while e is called identity if it is both right and left identity An element e ofH is called right scalar identity if x = x middot e for all x in H If x = e middot x for all x in H then eis called left scalar identity while e is called scalar identity if it is both right and left scalaridentity [4131888] When a left (resp right) scalar identity exists in H then it is unique

Example 8 The Cayley Tables 17 and 18 describe leftright almost-hypergroups with leftrightscalar identity

Table 17 Left almost-hypergroup with left scalar identity


( ) ( )= b a c c a b

ii In any right almost-hypergroup the left inverted associativity of the induced hypercompositions is valid

( ) ( )= b a c c a b


( ) ( ) | |b a c x H bx a c x H a bx c= isin cap ne empty = isin isin =

( ) ( )| | b x H a bx c x H a cx c a b= isin isin = isin cap ne empty =


( ) ( ) | |b a c x H b a xc x H a b xc= isin cap ne empty = isin isin =

( ) ( )| | b x H a c xb x H c a x c a b= isin isin = isin cap ne empty =


Corollary 9


( ) ( )= B A C C A B


( ) ( )= B A C C A B

5 Identities and Symmetric Elements Let H be a non-void set endowed with a hypercomposition An element e of H

is called right identity if isin sdotx x e for all x in H If isin sdotx e x for all x in H then e is called left identity while e is called identity if it is both right and left identity An element e of H is called right scalar identity if = sdotx x e for all x in H If = sdotx e x for all x in H then e is called left scalar identity while e is called scalar identity if it is both right and left scalar identity [4131888] When a left (resp right) scalar identity exists in H then it is unique

Example 8 The Cayley Tables 17 and 18 describe leftright almost-hypergroups with leftright scalar identity


1 2 3

1 1 2 3

2 3 123 13

3 2 12 123

Table 18 Right almost-hypergroup with right scalar identity

206




1 2 3

1 1 3 2

2 2 123 13

3 3 12 123

If the equality =e ee is valid for an identity e then e is called idempotent identity [43988] e is a right strong identity if isin sdot sube x x e e x for all x in H while e is a left

strong identity if isin sdot sube x e x e x e is a strong identity if it is right and left strong [432343988ndash90] Note that the strong identity needs not be unique [323439] Both the scalar identity and the strong identity are idempotent identities

Theorem 6 If e is a strong identity in a left almost-hypergroup H then

= = x e e x x for all isin minusx H e

Proof Suppose that isin y x e Then x ye y eisin sube Consequently =y x

Let e be an identity element in H and x an element in H Then x will be called right e-attractive if isin sdot e e x while it will be called left e-attractive if isin sdot e x e If x is both left and right e-attractive then it will be called e-attractive When there is no likelihood of confusion e can be omitted See [32] for the origin of the terminology

Proposition 14 In a left almost-hypergroup with idempotent identity e e e is the set of right e-attractive elements of H and e e is the set of left e-attractive elements of H

Proof Suppose that x is a right attractive element of H Then isine ex Thus isin x e e Additionally if isin x e e then isine ex Hence e e consists of all the right attractive elements of H The rest follows in a similar way

When the identity is strong and x is an attractive element then sdot = sdot = e x x e e x while if x is non-attractive then sdot = sdot =e x x e x is valid In the case of a strong identity the non-attractive elements are called canonical See [32] for the origin of the terminology

Proposition 15 If x is not an e-attractive element in a left almost-hypergroup with idempotent identity e then xe consists of elements that are not e-attractive

Proof Suppose that isiny xe and assume that y is attractive Then

( ) ( )sube = = ye xe e ee x ex

Since isin e ey it follows that isin e ex which is absurd

Proposition 16 If x is a right (resp left) e-attractive element in a transposition left almost-hypergroup with idempotent identity e then all the elements of xe are right (resp left) e-attractive

If the equality e = ee is valid for an identity e then e is called idempotent iden-tity [43988] e is a right strong identity if x isin x middot e sube e x for all x in H while e isa left strong identity if x isin e middot x sube e x e is a strong identity if it is right and leftstrong [432343988ndash90] Note that the strong identity needs not be unique [323439] Boththe scalar identity and the strong identity are idempotent identities


xe = ex = x for all x isin H minus e

Proof Suppose that y isin xe Then x isin ye sube y e Consequently y = x

Let e be an identity element in H and x an element in H Then x will be called righte-attractive if e isin e middot x while it will be called left e-attractive if e isin x middot e If x is both left andright e-attractive then it will be called e-attractive When there is no likelihood of confusione can be omitted See [32] for the origin of the terminology

Proposition 14 In a left almost-hypergroup with idempotent identity e ee is the set of righte-attractive elements of H and ee is the set of left e-attractive elements of H

Proof Suppose that x is a right attractive element of H Then e isin ex Thus x isin eeAdditionally if x isin ee then e isin ex Hence ee consists of all the right attractive elementsof H The rest follows in a similar way

When the identity is strong and x is an attractive element then e middot x = x middot e = e xwhile if x is non-attractive then e middot x = x middot e = x is valid In the case of a strong identitythe non-attractive elements are called canonical See [32] for the origin of the terminology

Proposition 15 If x is not an e-attractive element in a left almost-hypergroup with idempotentidentity e then xe consists of elements that are not e-attractive

Proof Suppose that y isin xe and assume that y is attractive Then

ye sube (xe)e = (ee)x = ex

Since e isin ey it follows that e isin ex which is absurd


Proof Let y isin xe then x isin ye Moreover x isin ee Thus ee cap ye 6= empty which impliesthat ee cap ey 6= empty Therefore e isin ey

207


Proposition 17 If x is a right (resp left) attractive element in a transposition left almost-hypergroup with idempotent identity e then its right (resp left) inverses are also right (resp left)attractive elements

Proof Since e isin ex it follows that x isin ee Moreover if xprime is a right inverse of x thene isin xxprime Therefore x isin exprime Consequently ee asymp exprime Transposition gives ee asymp exprime andsince e is idempotent e isin exprime Thus xprime is right attractive

Corollary 10 If x is not a right (resp left) attractive element in a transposition left almost-hypergroup with idempotent identity e then its right (resp left) inverses are also not right (respleft) attractive elements

Proposition 18 If a left almost-hypergroup (resp right almost-hypergroup) has a right scalaridentity (resp left scalar identity) then it is a hypergroup

Proof Suppose that e is a right scalar identity in a left almost-hypergroup H Then for anytwo elements b c in H we have

bc = (be)c = (ce)b = cb

Hence H is commutative and therefore H is a hypergroup Similar is the proof for theright almost-hypergroups

Proposition 19 If a left almost-hypergroup or a right almost-hypergroup H has a strong identity eand if it consists only of attractive elements then x y isin xy

Proof (ex)y = e xy = e y cup xy and (yx)e = yx cup e Since (ex)y = (yx)e it derivesthat yx cup e = xy cup e y Similarly xy cup e = yx cup e x Therefore

xy cup e = yx cup e x = xy cup e y cup e x = xy cup e x y

Hence x y isin xy

Corollary 11 If a left almost-hypergroup or a right almost-hypergroup H has a strong identity eand if it consists of attractive elements only then

i x isin xy and x isin yx for all x y isin Hii xx = xx = H for all x y isin H

Proposition 20 If a left almost-hypergroup or a right almost-hypergroup H has a strong identity eand if the relation e isin bc implies that e isin cb for all b c isin H then H is a hypergroup

Proof For any two elements in H we have

cb sube e cb = (ec)b = (bc)e = e cup bc (i)

If e isin bc then cb sube bc Moreover

bc sube e bc = (eb)c = (cb)e = e cup cb (ii)

According to the assumption if e isin bc then e isin cb Hence bc sube cb Thus cb = bc Next ife isin bc then e isin cb and (i) implies that cb sube bc while (ii) implies that bc sube cb Thus cb = bcConsequently H is commutative and therefore H is a hypergroup

208


Proposition 21 If a left almost-hypergroup or a right almost-hypergroup H has a strong identity eand if A is the set of its attractive elements then

AA sube A and AA sube A

Proof Since A is the set of the attractive elements ee = A and ee = A is valid Therefore

AA = (ee)(ee) sube (eee)e = (ee)e

But ee = ee thus(ee)e = (ee)e = eee = ee = A

Consequently AA sube A Similarly AA sube A

An element xprime is called right e-inverse or right e-symmetric of x if there exists a rightidentity xprime 6= e such that e isin x middot xprime The definition of the left e-inverse or left e-symmetric isanalogous to the above while xprime is called e-inverse or e-symmetric of x if it is both right andleft inverse with regard to the same identity e If e is an identity in a left almost-hypergroupH then the set of the left inverses of x isin H with regard to e will be denoted by Sel(x) whileSer(x) will denote the set of the right inverses of x isin H with regard to e The intersectionSel(x) cap Ser(x) will be denoted by Se(x)

Proposition 22 If e is an identity in a left almost-hypergroup H then

Ser(x) = (ex) middot middote and Ser(x) = (xe) middot middote

Proof y isin ex if and only if e isin yx This means that either y isin Sel(x) or y = e if x is rightattractive Hence ex sube e cup Sel(x) The rest follows in a similar way

Corollary 12 If Sel(x) cap Ser(x) 6= empty x isin H then xe cap ex 6= empty

Proposition 23 If H is a transposition left almost-hypergroup with an identity e and z isin xy then

i ey cap xprimez 6= empty for all xprime isin Sel(x)ii xe cap zyprime 6= empty for all yprime isin Ser(y)

Proof z isin xy implies that x isin zy and that y isin xz Let xprime isin Sel(x) and yprime isin Ser(y)Then e isin xprimex and e isin yyprime Thus x isin xprimee and y isin eyprime Therefore xprimee cap zy 6= empty andxz cap eyprime 6= empty Hence because of the transposition ey cap xprimez 6= empty and xe cap zyprime 6= empty

Proposition 24 Let H be a transposition left almost-hypergroup with a strong identity e and letx y z be elements in H such that x y z 6= e and z isin xy Then

i if Sel(x) cap Sel(z) = empty then y isin xprimez for all xprime isin Sel(x)ii if Ser(y) cap Ser(z) = empty then x isin zyprime for all yprime isin Ser(y)

Proof (i) According to Proposition 23 z isin xy implies that ey cap xprimez 6= empty for all xprime isin Sl(x)Since e is strong ey = e y Hence e y cap xprimez 6= empty But Sel(x) and Sel(z) are disjointThus e isin xprimez therefore y isin xprimez Analogous is the proof of (ii)

6 Substructures of the LeftRight Almost-Hypergroups

There is a big variety of substructures in the hypergroups which is much biggerthan the one in the groups Analogous is the variety of the substructures which arerevealed here in the case of the leftright almost-hypergroups For the consistency ofthe terminology [413335591ndash99] the terms semisub-leftright almost-hypergroup sub-

209


leftright almost-hypergroup etc will be used in exactly the same way as the prefixessub- and semisub- are used in the cases of the groups and the hypergroups eg the termssubgroup subhypergroup are used instead of hypersubgroup etc The following researchis inspired by the methods and techniques used in [93ndash97]

Let H be a left almost-hypergroup Then

Definition 12 A non-empty subset K of H is called semisub-left-almost-hypergroup (semisub-LA-hypergroup) when it is stable under the hypercomposition ie when it has the property xy sube K forall x y isin K

Definition 13 A semisub-LA-hypergroup K of H is a sub-left-almost-hypergroup (sub-LA-hypergroup) of H if it satisfies the reproductivity ie if the equality xK = Kx = K is valid for allx isin K

Example 9 In the left almost-hypergroup which is described in the Cayley Table 19 5 6 7is a semisub-LA-hypergroup while 1 2 3 4 1 2 3 4 1 2 5 6 7 and 3 4 5 6 7 aresub-LA-hypergroup



Proof (i) According to Proposition 23 isinz xy implies that primecap ne emptyey x z for all

( )primeisin lx S x Since e is strong = ey e y Hence primecap ne empty e y x z But ( )elS x and

( )elS z are disjoint Thus primenotin e x z therefore primeisin y x z Analogous is the proof of (ii)

6 Substructures of the LeftRight Almost-Hypergroups There is a big variety of substructures in the hypergroups which is much bigger than

the one in the groups Analogous is the variety of the substructures which are revealed here in the case of the leftright almost-hypergroups For the consistency of the terminology [413335591ndash99] the terms semisub-leftright almost-hypergroup sub-leftright almost-hypergroup etc will be used in exactly the same way as the prefixes sub- and semisub- are used in the cases of the groups and the hypergroups eg the terms subgroup subhypergroup are used instead of hypersubgroup etc The following research is inspired by the methods and techniques used in [93ndash97]


Definition 12 A non-empty subset K of H is called semisub-left-almost-hypergroup (semisub-LA-hypergroup) when it is stable under the hypercomposition ie when it has the property subexy K for all isinx y K

Definition 13 A semisub-LA-hypergroup K of H is a sub-left-almost-hypergroup (sub-LA-hypergroup) of H if it satisfies the reproductivity ie if the equality = =xK Kx K is valid for all isinx K

Example 9 In the left almost-hypergroup which is described in the Cayley Table 19 567 is

a semisub-LA-hypergroup while 12 34 1234 12567 and 34567 are sub-LA-hypergroup


1 2 3 4 5 6 7 1 12 12 1234 1234 12567 12567 12567 2 1 12 1234 1234 12567 12567 12567 3 1234 1234 34 4 34567 34567 34567 4 1234 1234 3 34 34567 34567 34567 5 12567 12567 34567 34567 57 7 67 6 12567 12567 34567 34567 67 7 7 7 12567 12567 34567 34567 67 67 67

Proposition 25 If a semisub-LA-hypergroup K of H is stable under the induced hypercompositions then K is a sub-LA-hypergroup of H

Proof We have to prove the reproductivity Let isinx K Obviously subexK K and subeKx K Next let y be an element of K Then |x y z H y xz= isin isin is a subset of

H Therefore there exists an element isinz K such that isin subey xz xK Thus subeK xK Similarly subey x K yields subeK Kx

Proposition 26 If K is a sub-LA-hypergroup of H then ( )H K H K ssdot sdot sube sdot sdot and

( )H K s H Ksdot sdot sube sdot sdot for all isins K

Proposition 25 If a semisub-LA-hypergroup K of H is stable under the induced hypercompositionsthen K is a sub-LA-hypergroup of H

Proof We have to prove the reproductivity Let x isin K Obviously xK sube K and Kx sube KNext let y be an element of K Then xy = z isin H | y isin xz is a subset of H Thereforethere exists an element z isin K such that y isin xz sube xK Thus K sube xK Similarly yx sube Kyields K sube Kx

Proposition 26 If K is a sub-LA-hypergroup of H then H middot middotK sube (H middot middotK)s andH middot middotK sube s(H middot middotK) for all s isin K

Proof Let r be an element in H middot middotK which does not belong to (H middot middotK)s Because of thereproductive axiom r isin Hs and since r isin (H middot middotK)s r must be a member of Ks Thusr isin Ks sube KK = K This contradicts the assumption and so H middot middotK sube (H middot middotK)s The secondinclusion follows similarly

Proposition 27 If K is a sub-LA-hypergroup of H A sube K and B sube H then

i A(B cap K) sube AB cap K andii (B cap K)A sube BA cap K

Proof Let t isin A(B cap K) Then t isin ax with a isin A and x isin B cap K Since x lies in B cap K itderives that x isin B and x isin K Hence ax sube aB and ax sube aK = K Thus ax sube AB cap K andtherefore t isin AB cap K The second inclusion follows similarly

210


The next definition introduces the notion of the closed sub-LARA-hypergroupsThe significance of the closed subhypergroups which is mentioned in [413335591ndash9699]remains exactly the same in the case of the sub-LARA-hypergroups

Definition 14 A sub-LA-hypergroup K of H is called right closed in H if it is stable under theright induced hypercomposition that is if ab sube K for any two elements a and b in K SimilarlyK is called left closed if ba sube K for all a b isin K K is closed when it is both right and left closed

Proposition 28

i K is a right closed sub-LA-hypergroup of H if and only if xK cap K = empty for every x isin H middot middotKii K is a left closed sub-LA-hypergroup of H if and only if Kx cap K = empty for every x isin H middot middotKiii K is a closed sub-LA-hypergroup of H if and only if xK cap K = empty and Kx cap K = empty for

every x isin H middot middotK

Proof (i) Suppose that xK cap K 6= empty for some x isin H So there exist y z isin K such that y isin xzor equivalently x isin yz Since K is right closed yz sube K therefore x isin K Conversely nowLet y z isin K Then for every x isin yz we have that y isin xz or equivalently xK cap K 6= emptyConsequently x isin K thus yz sube K The proof of (ii) is similar while (iii) derives directlyfrom (i) and (ii)

Proposition 29

i A sub-LA-hypergroup K of H is right closed in H if and only if

(H middot middotK)s = H middot middotK for all s isin K

ii A sub-LA-hypergroup K of H is left closed in H if and only if

s(H middot middotK) = H middot middotK for all s isin K

iii A sub-LA-hypergroup K of H is closed in H if and only if

s(H middot middotK) = (H middot middotK)s = H middot middotK for all s isin K

Proof (i) Let K be right closed in H Suppose that z lies in H middot middotK and assume thatzs cap K 6= empty Then there exists an element y in K such that y isin zs or equivalently z isin ysTherefore z isin K which is absurd Hence (H middot middotK)s sube H middot middotK Next because of Proposition26 H middot middotK sube (H middot middotK)s and therefore H middot middotK = (H middot middotK)s Conversely now Suppose that(H middot middotK)s = H middot middotK for all s isin K Then (H middot middotK)s cap K = empty for all s isin K Hence x isin rs and sor isin xs for all x s isin K and r isin H middot middotK Therefore xs cap (H middot middotK) = empty which implies thatxs sube K Thus K is right closed in H (ii) follows in a similar way and (iii) is an obviousconsequence of (i) and (ii)

Proposition 30

i If K is a sub-LA-hypergroup of H and

Kx cap (H middot middotK)x = empty

for all x isin H then K is right closed in Hii If K is a sub-LA-hypergroup of H and

xK cap x(H middot middotK) = empty

for all x isin H then K is left closed in H

211


Proof (i) From the reproductive axiom we have

K cup (H middot middotK) = H = Hx = Kx cup (H middot middotK)x

According to the hypothesis Kx cap (H middot middotK)x = empty which implies that K cap (H middot middotK)x = emptywhen x isin K Therefore H = K cup (H middot middotK)x is a union of disjoint sets Thus(H middot middotK)x = H middot middotK So per Proposition 29 K is right closed in H Similar is the proofof (ii)

Proposition 31

i If K is a right closed sub-LA-hypergroup in H A sube K and B sube H then

(B cap K)A = BA cap K

ii If K is a left closed sub-LA-hypergroup in H A sube K and B sube H then

A(B cap K) = AB cap K

Proof (i) Let t isin BA cap K Since K is right closed for any element y in B middot middotK it is validthat yA cap K sube yK cap K = empty Hence t isin (B cap K)A cap K But (B cap K)A sube KK = K Thust isin (B cap K)A Therefore BAcap K sube (B cap K)A Next the inclusion becomes equality becauseof Proposition 27 (ii) derives in a similar way

Proposition 32


(B cap K)A = (BA) cap K



Proof (i) Since B cap K sube B it derives that (B cap K)A sube BA Moreover A sube K andBcapK sube K thus (B cap K)A sube K Hence (B cap K)A sube (BA)capK For the reverse inclusionnow suppose that x isin (BA) cap K Then there exist a isin A and b isin B such that x isin baor equivalently b isin ax Since ax sube K it derives that b isin K and so b isin B cap K Thereforeba sube (B cap K)A Thus x isin (B cap K)A Hence (BA) cap K sube (B cap K)A (ii) derives in asimilar way

Although the non-void intersection of two sub-LA-hypergroups is stable under thehypercomposition it is usually not a sub-LA-hypergroup since the reproductive axiom isnot always valid in it

Proposition 33 The non-void intersection of any two closed sub-LA-hypergroups of H is a closedsub-LA-hypergroup of H

Proof Let K M be two closed LA-subhypergroups of H and suppose that x y are twoelements in K capM Then xy sube K and xy sube M Therefore xy sube K capM Next since K Mare closed LA-subhypergroups of H xy and yx are subsets of K capM Thus because ofProposition 25 K capM is a closed LA-subhypergroup of H

Corollary 13 The set of the closed sub-LA-hypergroups of H which are containing a non-voidsubset of H is a complete lattice

212


Proposition 34 If K is a closed sub-LA-hypergroups of H and x isin K then

xK = Kx = K = Kx = xK

Proof Since K is closed xK sube K and Kx sube K Let y isin K Because of the reproduc-tivity x isin yK or equivalently y isin xK Therefore xK = K Moreover since K is a LA-subhypergroups of H yx sube K Thus y isin Kx So Kx = K The equalities K = Kx = xKfollow in a similar way

Corollary 14 In any left almost-hypergroup K is a closed sub-LA-hypergroup if and only if

KK = KK = K

Definition 15 A sub-LA-hypergroup M of H is called right invertible if xy capM 6= empty impliesyx capM 6= empty while it is called left invertible if yx capM 6= empty implies xy capM 6= empty If M isright and left invertible then it is called invertible

Direct consequence of the above definition is the following proposition

Proposition 35

i M is a right invertible sub-LA-hypergroup of H if and only if

x isin MyrArr y isin Mx x y isin H

ii M is a left invertible sub-LA-hypergroup of H if and only if

x isin yMrArr y isin xM x y isin H

Proposition 36 If K is an invertible sub-LA-hypergroup of H then K is closed

Proof Let x isin KK Then K cap xK 6= empty and xK cap K 6= empty Since K is invertibleKx cap K 6= empty Thus x isin KK But K is a LA-subhypergroup of H so KK = K There-fore x isin K Hence KK sube K Similarly KK sube K and so the Proposition

Definition 16 If H has an identity e then a sub-LA-hypergroup K of H is called symmetric ifx isin K implies Sel(x) cup Ser(x) sube K

Proposition 37 To any pair of symmetric sub-LA-hypergroups K and M of a LA-hypergroup Hthere exists a least symmetric sub-LA-hypergroup K orM containing them both

Proof Let U be the set of all symmetric subhypergroups R of H which contain both Kand M The intersection of all these symmetric subhypergroups R of H is a symmetricsubhypergroup with the desired property

7 Fortification in Transposition Left Almost-Hypergroups

The transposition left almost-hypergroups can be fortified through the introduction ofneutral elements Next we will present two such hypercompositional structures

Definition 17 A transposition LA-hypergroup H is left fortified if it contains an element e whichsatisfies the axioms

i e is a left identity and ee = eii for every x isin H middot middote there exists a unique y isin H middot middote such that e isin xy and e isin yx

213


For x isin H middot middote the notation xminus1 is used for the unique element of H middot middote thatsatisfies axiom (ii) Clearly

(xminus1)minus1

= x The next results are obvious

Proposition 38

i If x 6= e then ex = xii ee = H

Proposition 39 Let x isin H middot middote Then e isin xy or e isin yx implies y isin

xminus1 e

Now the role of the identity in the transposition left almost-hypergroups can be clari-fied

Proposition 40 The identity e of the transposition left almost-hypergroup is left strong

Proof It suffices to prove that ex sube e x For x = e the inclusion is valid Letx 6= e Suppose that y isin ex Then x isin ey But e isin xxminus1 hence x isin exminus1 Thus(ey) cap

(exminus1) 6= empty The transposition axiom gives e = ee = yxminus1 By the previous

proposition y isin e x Therefore the proposition holds

Proposition 41 The identity e of the transposition left almost-hypergroup is unique

Proof Suppose that u is an identity distinct from e Then there would exist the inverse ofe ie an element v distinct from u such that u isin ev which is absurd because ev sube e v

Proposition 42 If H consists of attractive elements only and e 6= x then

ex = exminus1 =

e xminus1= xe

Proof Since e is a left strong identity exminus1 =

e xminus1 is valid Moreover

ex = z isin H | e isin zx =

e xminus1

and xe = z isin H | e isin xz =

e xminus1

Corollary 15 If A is a non-empty subset of H and e isin A then

eA = eAminus1 = Aminus1 cup e = Ae

Definition 18 A transposition polysymmetrical left almost-hypergroup is a transposition leftalmost-hypergroup P that contains a left idempotent identity e which satisfies the axiom

for every x isin P middot middote there exists at least one element xprime isin P middot middote a symmetricof x such that e isin x xprime which furthermore satisfies e isin xprimex

The set of the symmetric elements of x is denoted by S(x) and it is called the symmetric set of x

Example 10 Cayley Table 20 describes a transposition polysymmetrical left almost-hypergroup inwhich the element 1 is a left idempotent identity

214


Table 20 Transposition polysymmetrical left almost-hypergroup


( ) ( )minuscap ne empty1 e y e x The transposition axiom gives minus= = 1e ee yx By the previous

proposition isin y e x Therefore the proposition holds


Proof Suppose that u is an identity distinct from e Then there would exist the inverse of e ie an element v distinct from u such that isinu ev which is absurd because

sube ev e v

Proposition 42 If H consists of attractive elements only and nee x then

minus minus= = =1 1 e x ex e x x e

Proof Since e is a left strong identity minus minus=1 1ex e x is valid Moreover

1 | e x z H e zx e xminus= isin isin = and 1 | x e z H e xz e xminus= isin isin =

Corollary 15 If A is a non-empty subset of H and notine A then

minus minus= = cup =1 1 e A eA A e A e

Definition 18 A transposition polysymmetrical left almost-hypergroup is a transposition left almost-hypergroup P that contains a left idempotent identity e which satisfies the axiom

for every x P eisin sdotsdot there exists at least one element acutex P eisin sdotsdot a symmetric

of x such that isin e x x and furthermore acutex satisfies isin acutee x x

The set of the symmetric elements of x is denoted by ( )S x and it is called the symmetric set of x

Example 10 Cayley Table 20 describes a transposition polysymmetrical left almost-hypergroup in which the element 1 is a left idempotent identity


1 2 3

1 1 12 13

2 13 123 13

3 12 12 123

Proposition 43 ex always contains the element x

Corollary 16 If X is non-empty then sube X e X

Proposition 44 Let nex e Then


Corollary 16 If X is non-empty then X sube eX

Proposition 44 Let x 6= e Then

i S(x) cup e = ex and S(x) = xe if x is attractiveii eS(x) = ex if e is left strong identity and x is attractiveiii S(x) = xe = ex if x is non attractive

Corollary 17 Let X be a non-empty set and e isin X Then

i S(X) cup e = eX and S(X) = Xe if X contains an attractive elementii eS(X) = eX if e is left strong identity and X contains an attractive elementiii S(X) = Xe = eX if X consists of non-attractive elements

Proposition 45 For every element x of a transposition polysymmetrical left almost-hypergroup itholds ex sube e cup S(S(x)) while for every xprime isin S(x) it holds (ex) cap S(xprime) 6= empty

Proof Let y 6= e and y isin ex then x isin ey Moreover for every xprime isin S(x) it holds x isin exprimeConsequently exprime cap ey 6= empty so per transposition axiom ee cap yxprime 6= empty that is e isin yxprime

and thus y isin S(xprime) sube S(S(x))

Proposition 46 If x 6= e is a right attractive element of a transposition polysymmetrical leftalmost-hypergroup then S(x) consists of left attractive elements

Proof Let e isin ex Then x isin ee Moreover if xprime is an arbitrary element from S(x) thene isin xxprime Therefore x isin exprime Consequently ee cap exprime 6= empty Per transposition axiomee cap xprimee 6= empty So e isin xprimee Thus xprime is an attractive element

Corollary 18 If x is a non-attractive element then S(x) consists of non-attractive elements only

Corollary 19 If x 6= e is an attractive element of a transposition polysymmetrical left almost-hypergroup then S(x) consists of attractive elements only

Proposition 47

i If x is a right attractive element of a transposition polysymmetrical left almost-hypergroupthen all the elements of xe are right attractive

ii If x is a left attractive element of a transposition polysymmetrical left almost-hypergroupthen all the elements of ex are left attractive

Proof Assuming that x is a right attractive element we have that e isin ex which implies thatx isin ee Additionally if z is an element in xe then x isin ze Thus (ze) cap (ee) 6= empty Per

215


transposition axiom (ee) cap (ez) 6= empty and therefore e isin ez ie z is right attractive Similaris the proof of (ii)


In [7093] and later in [4] with more details it was proved that the group can bedefined with the use of two axioms only the associativity and the reproductivity Likewisethe leftright almost-group is defined here and their existence is proved via examples Thestudy of this structure reveals a very interesting research area in abstract algebra

This paper focuses on the study of the more general structure ie of the leftrightalmost-hypergroup The enumeration of these structures showed that they appear morefrequently than the hypergroups Indeed in the case of the hypergroupoids with threeelements there exists one hypergroup in every 1740 hypergroupoids while there is onenon-trivial purely left almost-hypergroup in every 612 hypergroupoids The same holdsfor the non-trivial purely right almost-hypergroups as it is proved in this paper that thecardinal number of the left almost-hypergroups is equal to the cardinal number of theright almost-hypergroups over a set E Moreover there is one non-trivial left and rightalmost-hypergroup in every 5735 hypergroupoids Considering the trivial cases as wellie left and right almost-hypergroups which are also non-commutative hypergroups thereexists one left almost-hypergroup in every 453 hypergroupoids This frequency whichis nearly 4 times higher than that of the hypergroups justifies a more thorough study ofthese structures

Subsequently these structures were equipped with more axioms the first one of whichis the transposition axiom


The transposition leftright almost-hypergroup is studied hereIn [4] though the reverse transposition axiom was introduced


The study of the reverse transposition leftright almost-hypergroup is an open problemAdditionally open problems for algebraic research are the studies of the properties of

all the structures which are introduced in this paper (weak leftright almost-hypergroupleftright almost commutative hypergroup join leftright almost hypergroup reverse joinleftright almost hypergroup weak leftright almost commutative hypergroup) as wellas their enumerations Especially for the enumeration problems it is worth mentioningthose which are associated with the rigid hypercompositional structures that is hyper-compositional structures whose automorphism group is of order 1 The conjecture is thatthere exists only one rigid left almost-hypergroup (and one rigid right almost-hypergroup)while it is known that there exist six such hypergroups five of which are transpositionhypergroups [81]

Author Contributions Conceptualization investigation CGM and NY software validationwritingmdashoriginal draft preparation writingmdashreview and editing funding acquisition CGM Bothauthors have read and agreed to the published version of the manuscript

Funding This research received no external funding The APC was funded by the MDPI journalMathematics and the National and Kapodistrian University of Athens





216


Appendix A

The following is the Mathematica [100] package that implements the axioms of leftinverted associativity right inverted associativity associativity and reproductivity fortesting whether a magma is a LA-hypergroup a RA-hypergroup a LRA-hypergroup or anon-commutative hypergroup

BeginPackage[LRHtestlsquo]Clear[LRHtestlsquo]

LRHtestusage = LRHtest[groupoid] returns1 for Left Asso2 for Right Asso3 for Left+Right Asso4 for Hypergroup5 for Hypergroup + Left Asso6 for Hypergroup + Right Asso7 for Hypergroup + Left+Right Asso

Begin[lsquoPrivatelsquo]Clear[LRHtestlsquoPrivatelsquo]

LRHtest[groupoid_List] = Module[r r = 0If[groupoid = Transpose[groupoid] ampamp ReproductivityTest[groupoid]If[LeftAs[groupoid] r = 1]If[RightAs[groupoid] r = r + 2]If[Asso[groupoid] r = r + 4]] Return[r]]

LeftAs[groupoid_List] =Not[MemberQ[Flatten[Table[Union[Flatten[Union[Extract[groupoid

Distribute[groupoid[[i j]] k List]]]]] ==Union[Flatten[Union[Extract[groupoid

Distribute[groupoid[[k j]] i List]]]]] i 1Length[groupoid] j 1 Length[groupoid] k 1Length[groupoid]] 2] False]]

RightAs[groupoid_List] =Not[MemberQ[Flatten[Table[Union[Flatten[

Union[Extract[groupoidDistribute[i groupoid[[j k]] List]]]]] ==

Union[Flatten[Union[Extract[groupoid

Distribute[k groupoid[[j i]] List]]]]] i 1Length[groupoid] j 1 Length[groupoid] k 1Length[groupoid]] 2] False]]

Asso[groupoid1_List] =

217


Not[MemberQ[Flatten[Table[Union[Flatten[

Union[Extract[groupoid1Distribute[groupoid1[[i j]] k List]]]]] ==

Union[Flatten[Union[Extract[groupoid1

Distribute[i groupoid1[[j k]] List]]]]] i 1Length[groupoid1] j 1 Length[groupoid1] k 1Length[groupoid1]] 2] False]]

ReproductivityTest[groupoid_List] =Min[Table[

Length[Union[Flatten[Transpose[groupoid][[j]]]]] j 1Length[groupoid]]] == Length[groupoid] ampamp

Min[Table[Length[Union[Flatten[groupoid[[j]]]]] j 1Length[groupoid]]] == Length[groupoid]

End[]EndPackage[]

Use of the packagefor checking a magma for instance the following one

1 1 231 1 23123 23 13

write in Mathematica

In[1]=LRHtest[1 1 2 3 1 1 2 3 1 2 3 23 1 3]

And the output is

Out[1]=2

where number 2 corresponds to laquoRA-hypergroupraquo

References1 Marty F Sur une Geacuteneacuteralisation de la notion de Groupe Huitiegraveme Congregraves des Matheacutematiciens Scand Stockholm 1934 45ndash492 Marty F Rocircle de la notion drsquohypergroupe dans lrsquoeacutetude des groupes non abeacuteliens Comptes Rendus Acad Sci Paris 1935 201

636ndash6383 Marty F Sur les groupes et hypergroupes attacheacutes agrave une fraction rationnelle Ann Sci lEacutecole Norm Supeacuterieure 1936 53 83ndash123

[CrossRef]4 Massouros C Massouros G An Overview of the Foundations of the Hypergroup Theory Mathematics 2021 9 1014 [CrossRef]5 Kazim MA Naseeruddin M On almost semigroups Portugaliae Math 1977 36 41ndash476 Aslam M Aroob T Yaqoob N On cubic Γ-hyperideals in left almost Γ-semihypergroups Ann Fuzzy Math Inform 2013 5

169ndash1827 Gulistan M Yaqoob N Shahzad M A Note on Hv-LA-Semigroups UPB Sci Bull 2015 77 93ndash1068 Yaqoob N Aslam M Faisal On soft Γ-hyperideals over left almost Γ-semihypergroups J Adv Res Dyn Control Syst 2012 4

1ndash129 Yaqoob N Corsini P Yousafzai F On intra-regular left almost semi-hypergroups with pure left identity J Math 2013

[CrossRef]

218


10 Yaqoob N Cristea I Gulistan M Left almost polygroups Ital J Pure Appl Math 2018 39 465ndash47411 Amjad V Hila K Yousafzai F Generalized hyperideals in locally associative left almost semihypergroups N Y J Math 2014

20 1063ndash107612 Hila K Dine J On hyperideals in left almost semihypergroups ISRN Algebra 2011 [CrossRef]13 Corsini P Prolegomena of Hypergroup Theory Aviani Editore Tricesimo Udine Italy 199314 Corsini P Leoreanu V Applications of Hyperstructures Theory Kluwer Academic Publishers Dordrecht The Netherlands 200315 Krasner M Hypergroupes moduliformes et extramoduliformes Comptes Rendus Acad Sci Paris 1944 219 473ndash47616 Mittas J Hypergroupes et hyperanneaux polysymetriques Comptes Rendus Acad Sci Paris 1970 271 920ndash92317 Mittas J Hypergroupes canoniques hypervalues Comptes Rendus Acad Sci Paris 1970 271 4ndash718 Mittas J Hypergroupes canoniques Math Balk 1972 2 165ndash17919 Mittas J Hypergroupes canoniques values et hypervalues Hypergroupes fortement et superieurement canoniques Bull Greek

Math Soc 1982 23 55ndash8820 Mittas J Hypergroupes polysymetriques canoniques In Proceedings of the Atti del Convegno su Ipergruppi Altre Strutture

Multivoche e loro Applicazioni Udine Italy 15ndash18 October 1985 pp 1ndash2521 Mittas J Generalized M-Polysymmetric Hypergroups In Proceedings of the 10th International Congress on Algebraic Hyper-

structures and Applications Brno Czech Republic 3ndash9 September 2008 pp 303ndash30922 Massouros CG Mittas J On the theory of generalized M-polysymmetric hypergroups In Proceedings of the 10th International

Congress on Algebraic Hyperstructures and Applications Brno Czech Republic 3ndash9 September 2008 pp 217ndash22823 Massouros CG Canonical and Join Hypergroups An Stiintifice Univ Alexandru Ioan Cuza 1996 42 175ndash18624 Yatras C M-polysymmetrical hypergroups Riv Mat Pura Appl 1992 11 81ndash9225 Yatras C Types of Polysymmetrical Hypergroups AIP Conf Proc 2018 1978 340004-1ndash340004-5 [CrossRef]26 Bonansinga P Sugli ipergruppi quasicanonici Atti Soc Pelor Sci Fis Mat Nat 1981 27 9ndash1727 Massouros CG Quasicanonical hypergroups In Proceedings of the 4th International Congress on Algebraic Hyperstructures

and Applications Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991 pp 129ndash13628 Comer S Polygroups derived from cogroups J Algebra 1984 89 397ndash405 [CrossRef]29 Davvaz B Polygroup Theory and Related Systems World Scientific Singapore 201330 Massouros GG Zafiropoulos FA Massouros CG Transposition polysymmetrical hypergroups In Proceedings of the 8th

International Congress on Algebraic Hyperstructures and Applications Samothraki Greece 1ndash9 September 2002 Spanidis PressXanthi Greece 2003 pp 191ndash202

31 Massouros GG Fortified join hypergroups and join hyperrings An Stiintifice Univ Alexandru Ioan Cuza Sect I Mat 1995 XLI37ndash44

32 Massouros GG Massouros CG Mittas J Fortified join hypergroups Ann Math Blaise Pascal 1996 3 155ndash169 [CrossRef]33 Jantosciak J Transposition hypergroups Noncommutative Join Spaces J Algebra 1997 187 97ndash119 [CrossRef]34 Jantosciak J Massouros CG Strong Identities and fortification in Transposition hypergroups J Discret Math Sci Cryptogr

2003 6 169ndash193 [CrossRef]35 Massouros GG Hypercompositional structures in the theory of languages and automata An Stiintifice Univ Alexandru Ioan

Cuza Sect Inform 1994 III 65ndash7336 Massouros GG Hypercompositional structures from the computer theory Ratio Math 1999 13 37ndash4237 Massouros GG On the attached hypergroups of the order of an automaton J Discret Math Sci Cryptogr 2003 6 207ndash215

[CrossRef]38 Massouros CG Massouros GG Hypergroups associated with graphs and automata AIP Conf Proc 2009 1168 164ndash167

[CrossRef]39 Massouros CG Massouros GG On the algebraic structure of transposition hypergroups with idempotent identity Iran J

Math Sci Inform 2013 8 57ndash7440 Ameri R Topological transposition hypergroups Ital J Pure Appl Math 2003 13 181ndash18641 De Salvo M Lo Faro G On the n-complete hypergroups Discret Math 1999 208ndash209 177ndash188 [CrossRef]42 De Salvo M Freni D Lo Faro G A new family of hypergroups and hypergroups of type U on the right of size five Far East J

Math Sci 2007 26 393ndash41843 De Salvo M Freni D Lo Faro G Fully simple semihypergroups J Algebra 2014 399 358ndash377 [CrossRef]44 De Salvo M Fasino D Freni D Lo Faro G 1-hypergroups of small size Mathematics 2021 9 108 [CrossRef]45 Gutan M Sur une classe drsquohypergroupes de type C Ann Math Blaise Pascal 1994 1 1ndash19 [CrossRef]46 Kankaras M Cristea I Fuzzy reduced hypergroups Mathematics 2020 8 263 [CrossRef]47 Cristea I Complete hypergroups 1-hypergroups and fuzzy sets An St Univ Ovidius Constanta 2000 10 25ndash3848 Angheluta C Cristea I On Atanassovrsquos intuitionistic fuzzy grade of complete hypergroups J Mult Val Log Soft Comput 2013

20 55ndash7449 Massouros CG Massouros GG On certain fundamental properties of hypergroups and fuzzy hypergroupsmdashMimic fuzzy

hypergroups Intern J Risk Theory 2012 2 71ndash8250 Hoskova-Mayerova S Al Tahan M Davvaz B Fuzzy multi-hypergroups Mathematics 2020 8 244 [CrossRef]51 Al Tahan M Hoskova-Mayerova S Davvaz B Fuzzy multi-polygroups J Intell Fuzzy Syst 2020 38 2337ndash2345 [CrossRef]

219


52 Chvalina J Novaacutek M Smetana B Stanek D Sequences of Groups Hypergroups and Automata of Linear Ordinary DifferentialOperators Mathematics 2021 9 319 [CrossRef]

53 Novaacutek M Krehliacutek S Cristea I Cyclicity in EL-hypergroups Symmetry 2018 10 611 [CrossRef]54 Novaacutek M Krehliacutek Š ELndashhyperstructures revisited Soft Comput 2018 22 7269ndash7280 [CrossRef]55 Massouros GG On the Hypergroup Theory Fuzzy Syst AI Rep Lett Acad Romana 1995 IV 13ndash2556 Prenowitz W Projective Geometries as multigroups Am J Math 1943 65 235ndash256 [CrossRef]57 Prenowitz W Descriptive Geometries as multigroups Trans Am Math Soc 1946 59 333ndash380 [CrossRef]58 Prenowitz W Spherical Geometries and mutigroups Can J Math 1950 2 100ndash119 [CrossRef]59 Prenowitz W A Contemporary Approach to Classical Geometry Am Math Month 1961 68 1ndash67 [CrossRef]60 Prenowitz W Jantosciak J Geometries and Join Spaces J Reine Angew Math 1972 257 100ndash12861 Prenowitz W Jantosciak J Join Geometries A Theory of Convex Sets and Linear Geometry Springer BerlinHeidelberg Ger-

many 197962 Jantosciak J Classical geometries as hypergroups In Proceedings of the Atti del Convegno su Ipergruppi altre Structure

Multivoche et loro Applicazioni Udine Italy 15ndash18 October 1985 pp 93ndash10463 Jantosciak J A brif survey of the theory of join spaces In Proceedings of the 5th International Congress on Algebraic Hyperstruc-

tures and Applications Iasi Rumania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA 1994 pp 109ndash12264 Barlotti A Strambach K Multigroups and the foundations of Geometry Rend Circ Mat Palermo 1991 XL 5ndash68 [CrossRef]65 Freni D Sur les hypergroupes cambistes Rend Ist Lomb 1985 119 175ndash18666 Freni D Sur la theacuteorie de la dimension dans les hypergroupes Acta Univ Carol Math Phys 1986 27 67ndash8067 Massouros CG Hypergroups and convexity Riv Mat Pura Appl 1989 4 7ndash2668 Mittas J Massouros CG Hypergroups defined from linear spaces Bull Greek Math Soc 1989 30 63ndash7869 Massouros CG Hypergroups and Geometry Mem Acad Romana Math Spec Issue 1996 XIX 185ndash19170 Massouros CG On connections between vector spaces and hypercompositional structures Ital J Pure Appl Math 2015 34

133ndash15071 Massouros GG Massouros CG Hypercompositional algebra computer science and geometry Mathematics 2020 8 1338

[CrossRef]72 Dramalidis A On geometrical hyperstructures of finite order Ratio Math 2011 21 43ndash5873 Massouros GG Mittas J LanguagesmdashAutomata and hypercompositional structures In Proceedings of the 5th International

Congress on Algebraic Hyperstructures and Applications Xanthi Greece 27ndash30 June 1990 World Scientific Singapore 1991 pp137ndash147

74 Massouros GG Automata and hypermoduloids In Proceedings of the 5th International Congress on Algebraic Hyperstructuresand Applications Iasi Rumania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA 1994 pp 251ndash265

75 Massouros CG Massouros GG The transposition axiom in hypercompositional structures Ratio Math 2011 21 75ndash9076 Massouros CG Dramalidis A Transposition Hv-groups ARS Comb 2012 106 143ndash16077 Vougiouklis T Hyperstructures and Their Representations Hadronic Press Palm Harbor FL USA 199478 Tsitouras CG Massouros CG On enumeration of hypergroups of order 3 Comput Math Appl 2010 59 519ndash523 [CrossRef]79 Massouros CG Tsitouras CG Enumeration of hypercompositional structures defined by binary relations Ital J Pure Appl


J Comb 2012 33 1777ndash1786 [CrossRef]81 Massouros CG On the enumeration of rigid hypercompositional structures AIP Conf Proc 2014 1648 740005ndash1ndash740005ndash6

[CrossRef]82 Massouros CG Massouros GG On 2-element Fuzzy and Mimic Fuzzy Hypergroups AIP Conf Proc 2012 1479 2213ndash2216

[CrossRef]83 Cristea I Jafarpour M Mousavi S Soleymani A Enumeration of Rosenberg hypergroups Comput Math Appl 2010 60

2753ndash2763 [CrossRef]84 Jafarpour M Cristea I Tavakoli A A method to compute the number of regular reversible Rosenberg hypergroup ARS Comb

2018 128 309ndash32985 Bayon R Lygeros N Advanced results in enumeration of hyperstructures J Algebra 2008 320 821ndash835 [CrossRef]86 Nordo G An algorithm on number of isomorphism classes of hypergroups of order 3 Ital J Pure Appl Math 1997 2 37ndash4287 Ameri R Eyvazi M Hoskova-Mayerova S Advanced results in enumeration of hyperfields AIMS Math 2020 5 6552ndash6579

[CrossRef]88 Massouros CG Massouros GG Identities in multivalued algebraic structures AIP Conf Proc 2010 1281 2065ndash2068 [CrossRef]89 Massouros CG Massouros GG Transposition hypergroups with idempotent identity Int J Algebr Hyperstruct Appl 2014 1

15ndash2790 Massouros CG Massouros GG Transposition polysymmetrical hypergroups with strong identity J Basic Sci 2008 4 85ndash9391 Krasner M La loi de JordanmdashHolder dans les hypergroupes et les suites geacuteneacuteratrices des corps de nombres PmdashAdiqes (I) Duke

Math J 1940 6 120ndash140 [CrossRef] (II) Duke Math J 1940 7 121ndash135 [CrossRef]92 Krasner M Kuntzmann J Remarques sur les hypergroupes Comptes Rendus Acad Sci Paris 1947 224 525ndash527

220


93 Massouros CG On the semi-subhypergroups of a hypergroup Int J Math Math Sci 1991 14 293ndash304 [CrossRef]94 Massouros GG The subhypergroups of the fortified join hypergroup Ital J Pure Appl Math 1997 2 51ndash6395 Massouros CG Massouros GG On subhypergroups of fortified transposition hypergroups AIP Conf Proc 2013 1558

2055ndash2058 [CrossRef]96 Massouros CG Some properties of certain subhypergroups Ratio Math 2013 25 67ndash7697 Yatras C Subhypergroups of M-polysymmetrical hypergroups In Proceedings of the 5th International Congress on Algebraic

Hyperstructures and Applications Iasi Rumania 4ndash10 July 1993 Hadronic Press Palm Harbor FL USA 1994 pp 123ndash13298 De Salvo M Freni D LoFaro G Hypercyclic subhypergroups of finite fully simple semihypergroups J Mult Valued Log Soft

Comput 2017 29 595ndash61799 Sureau Y Sous-hypergroupe engendre par deux sous-hypergroupes et sous-hypergroupe ultra-clos drsquoun hypergroupe Comptes

Rendus Acad Sci Paris 1977 284 983ndash984100 Wolfram S The Mathematica Book 5th ed Wolfram Media Champaign IL USA 2003

221

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Contents














v

About the Editor











vii



Algebra

























Stockholm in 1934















ix





















Editor

x

mathematics

Review















iations








40)






1 Introduction



1





2


2 Magma








(x y)rarr x y


(x y)rarr E





2 Magma















X Y X Y or X Y X Y



x y x y


x y E






these two elements





segment ab

Figure 1 Example 4


3




perp 1 2 3 4

1 1 3 2 32 3 2 1 43 1 2 1 14 2 3 4 1

perp 1 2 3 4

1 123 13 24 1342 3 1234 13 243 123 12 1234 14 23 23 34 13













4




ab =

ab i f b 6= 0


Q i f b = 0 a = 0









0

0 0

0 0

aif b

b

a bif b a

if b a





Figure 2 Example 7

Example 8 Let E





a b a if a ba b

E if a b

and

b a a if a bb a

E if a b









Figure 2 Example 7




a b =


and ba =









5








x y z x y z


associative magma









Figure 3 Example 10






c bb aa c







Figure 3 Example 10









hence (ba)c = b(ac)




6




for all a b isin E

















xperpy asymp yperpx






7












xperpE = Eperpx = E



perp 1 2 3 perp 1 2 3

1 1 2 3 1 13 3 22 2 3 1 2 2 123 23 3 1 2 3 2 1 123




8




perp a

a a


perp a b

a a bb b a






perp a b perp a b



perp a b c






9





















for all b isin G



10


for all b isin G





(aprimex)















= a


11











xa = b and ay = b





Or else


xa = b and ay = b



a b = b a


12













which is absurd










13

















14















15



(sprime primes)














(see


16

















xy = x(








17







x + y =












18











yminus1




= aaminus1






19






















20




xy =






is attractive









x + y =





21















22










for all a in K







23








cupkisinN

ak a isin H







24

















25














tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H





















26






















27
















Lx = (xL)L =(

xLminus1)


















Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot



(Lx)












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot




28














Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot







E3J

)L



Figure 4 Example 26


























Figure 4 Example 26





29




















30





given by

xlarrM

=

M i f x isin M


M=

M i f x isin M

Mx i f x isin M


xM =



P M =

xlarrM| x isin P

P M =

xrarr

M| x isin P

and P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M = xM | x isin P


= xrarrM

= xM Therefore

P M = P M = P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M


i x isin xlarrM

x isin xrarrM

and x isin xM


Msube xM

iii xM =(

xlarrM

)rarrM

=(

xrarrM

)larrM





i xlarrM

M = xM = xlarrMcupM

ii MxrarrM

= Mx = xrarrMcupM


xlarrM


Duality gives (ii)


QlarrM

M = QM = QlarrM

M cupM and MQrarrM

= MQ = QrarrMcupM


MxM = MxlarrM


M = xM M

31



xlarrMcupM

)= Mxlarr

McupMM = Mxlarr

McupM and



MM


MxM = M(

xlarrM

)rarrM

= MxlarrM

=(

xlarrM




(xy)larrM

M sube xlarrM


MM sube xrarr

Myrarr

McupM




xlarrM(yM) = xlarr

M

(yrarr

McupM

)= xlarr

Myrarr

Mcup xlarr

MM = xlarr

Myrarr

Mcup(

xlarrMcupM

)=(

xlarrM

yrarrMcup xlarr

M

)cupM


yrarrMcup xlarr

M= xlarr

Myrarr


(xlarr

Myrarr

Mcup xlarr

M

)cupM = xlarr

Myrarr

McupM



(XY)larrM

M sube XlarrM


MM sube Xrarr

MYrarr

McupM


(xy)M = xMyM cupM


(xy)M =((xy)larr

M

)rarrMsube(

xlarrM

ylarrMcupM

)rarrM

=(

xlarrM

ylarrM

)rarrMcupMrarr

Msube

sube(

xlarrM

)rarrM

(xlarr

M



(XY)M = XMYM cupM









tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M Thus in P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H















xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot



32














Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


are denoted by


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













and


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













then

xM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence

























Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot



and yM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence

























Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot





M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













yM 6= empty and yM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence















P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)







tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H
















P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)


(xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM)












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


zM = xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


(yM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


zM)







P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)



M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













xM cap zM


M M M M M M M M M M

M M M M M M M


x y z z M x y z M


valid Therefore







Therefore y x z w

for some My y

and Mw w

Thus xw y z

and so

M M M Mx w z y



pergroup


P M Hence













wM 6= empty Then

pM | xM isin yM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot














Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot




xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


wM cap zM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


yM 6= empty



P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is a transposition

hypergroup













Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


xM = xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot



in P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M Hence







tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M

33


Proposition 48 P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H















Since xM yM sube xM












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot








tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H
















P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is closed













Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot








tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M



P






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot


)is a transposition



(T






tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H














M












Lx = (xL)L =(

xLminus1)














E3J

)L




n that i


middot

replace the dot


middot

middot

replace the dot













tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H





















tion is open





a Kb K Ka KK a Ka




valid for all a b H



for each c H


c H






a b H

















34




















35

















36





















37




















38




















39

















40









41

mathematics

Article








1 Introduction





43













n=0

A1n




g(a) = (a) a isin A


44



⋃(ab)isinAtimesB














45


















46







a K = K a = aK = K = Ka = K a = a K








47



x y





x + y =x y




x y = y x =

H i f x = yx y

i f x y and x = e

x i f x y and x e







48








Is =x isin Σ

∣∣∣ δ(so x) = s

and Ps =




Proof Pr =si isin S





s + q = Ps







uisinPqcupPr

Pu)


⋃visinPscupPq

Pv)cup Pr

Thus(

⋃

visinPscupPq



s q = q s =

S i f s isin Pq






49



s q = q s =

S i f s isin Pq

empty i f s lt Pq





Proof Sincex y






50




vi v j


vertex v j











yminus1


Theorem 1 [57]






51


















x + y =x y

i f y minusx









e isin z(xxminus1


(xxminus1

)minus1

52




(xxminus1

)minus1implies that

(xxminus1

)minus1 sube[(

xxminus1)minus1






xn =


(1)

and

xm middot xn =

xm+n i f mn gt 0



(2)



M(x) =⋃

(mn)isinZtimesN0


)n


M(x) =⋃

(kl)isinN2

xk middot(xminus1

)l=

⋃

(kl)isinN2

xk middot xminusl



is established


53



M(x) =⋃

(mn)isinZ2


)n

Remark 1








)n





)n









(x middot xminus1



)minusn=

(x middot xminus1

)n





54






ande lt xm1minusm2

(xxminus1

)min|m1 | |m2 | i f m1m2 gt 0

Thuse lt xm middot

(x middot xminus1





)n


)s




)rit derives that


)kr= xm middot

(x middot xminus1

)n


(x middot xminus1





)n= xkp+r middot

(x middot xminus1


)n





)n=

(xminus1 middot x


)n=

(x middot xminus1

)p+n

Thusxkp sube

(x middot xminus1

)minusk(p+n)=

(x middot xminus1

)| k|(p+n)

and thereforee isin

(x middot xminus1


)n= xr middot

(x middot xminus1

)| k|(p+n)+n


(x middot xminus1


55






)




(x middot xminus1




(x middot xminus1

)n


22 The Hyperringoid












56








x + y =

x i f x = y]min

x y

max

x y

[ i f x y


x + y = ]minx y

max

x y

[ = y + x


x y z

max

x y z

[ = x + (y + z)


x y = y x =



x middot (y + z) =


ylttltzt = ⋃



x + y = [minx y

max

x y







57



A1 =

[1 00 0

] A2 =

[0 10 0

] A3 =

[0 01 0

] A4 =

[0 00 1

]

B1 =

[1 10 0

] B2 =

[0 10 1

] B3 =

[0 01 1

] B4 =

[1 01 0

] B5 =

[1 00 1

] B6 =

[0 11 0

]

C1 =

[0 11 1

] C2 =

[1 01 1

] C3 =

[1 11 0

] C4 =

[1 10 1

]

D1 =

[1 11 1

]

E1 =

[0 00 0

]









(x 0) + (y 0) =(x 0) (y 0)

(0 x) + (0 y) =(0 x) (0 y)

(x 0) + (0 y) = (0 y) + (x 0) =(x 0) (0 y)

f or x y

(x 0) + (0 x) = (0 x) + (x 0) =(x 0) (0 x) (0 0)

58



(x1 y1) middot (x2 y2) = (x1x2 y1y2)








ldquoHιτ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










σθω











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y





















( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










oυ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










π











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










π











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ν σηmicroε


Article







1 Introduction





oν ε


Article







1 Introduction





θε


Article







1 Introduction
















( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ν











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










γαγε


Article







1 Introduction





νrdquo [121]



ldquoKα


Article







1 Introduction





πεπερασmicro











hypercomposition






1 1





νην ε


Article







1 Introduction





θε


Article







1 Introduction





αν κατ











hypercomposition






1 1





τoacute συνεχ











hypercomposition






1 1





ς











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










πrsquo ε


Article







1 Introduction





θε











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ας











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










κβαλε


Article







1 Introduction





νrdquo [121]



ldquoT











hypercomposition






1 1





τ


Article







1 Introduction





α


Article







1 Introduction





τ


Article







1 Introduction






Article







1 Introduction





σα κα


Article







1 Introduction















( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










λλ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










λoις











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










στ











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y










ν


Article







1 Introduction





σαrdquo [121]




xy =κx + λy

∣∣∣ κλ isin F+ κ+ λ = 1



59
















60














x isin langAcup

yrang





61



















62









x + y =

max

x y

i f x y





x middot y = xyG

andx + y =

zG





0 + 0 = 0 0 + 1 = 1 + 0 = 1 1 + 1 = 0 1


63














x y





(x i) (xminusi) 0









64










x y













65








(v1 w1) + (v2 w2) =⋃


a(v w) = (av aw)




(a + b)(v w) =⋃


=

⋃(cv cw)


On the other hand

a(v w) + b(v w) = (av aw) + (bv bw) =⋃


=

=⋃


=

⋃(sv rw)






66








x












+ = isin = + isin


ie x y+












y =w isinM


ie x












+ = isin = + isin


ie x y+
























+ = isin = + isin


ie x y+












)





0 a



Fminus 0 F+






67








References















68
























69

























70
























71













121 Eυκλε











( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y





















( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) + =0 0 0 0x y x y

( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 0 0 0x y y x x y+ = + = for x yne

( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + =0 0 0 0 0 0 00x x x x x x




( ) ( ) ( )sdot =1 1 2 2 1 2 1 2 x y x y x x y y




















72









73

mathematics

Article







1 Introduction



75







76






and inseparable












77






















78



and a2 sim f i a3





|x+a| 1

|x+a+1| 1






micro(u) =

sum(xy)isinQ(u)

1|x y|

q(u) (1)





micro(u) =|H|2 1

|H||H|2

=1|H|



79





A capn

prodi=1

ai 6= emptyrArrn

prodi=1

ai sube A












80







micro(x) =1|Agx |








81















xminus1 u


H 0 1 2 30 0 1 2 31 1 2 0 3 12 2 0 3 1 23 3 1 2 0



82















1|x y| = sum

xisinH

1|x xminus1| =

sumaisinSc(H)

1|a aminus1| + sum

x 6isinSc(H)


x 6isinSc(H)

1|x xminus1|



1|x xminus1|







83


















bβ A

84





(bj)= 1










References







7 902 [CrossRef]

85













1479 2213ndash2216


86

mathematics

Article














nal affiliations








40)






1 Introduction



87








2 Preliminaries







88




















89






















90
















91






















(i) I sube P

92






sumj=1


Proof Let


sumj=1







+ 0 1 20 0 1 21 1 R 12 2 1 M

middot 0 1 20 0 0 01 0 1 22 0 2 0



1 an22 and



93





Ie = Qe1 capQe

2 cap capQen

with r(Qei ) = Pe







2 Pen




ass RI(

JI) = P

I P isin assR J


ht RI

PIle htRP le ht R

I

PI+ n









(a1a2at)


94



(a1 a2 at at+1 ad)

(a1 a2 at)



a1 a2 at at+1 ad















(0 R 2) = r isin R | r middot 2 = 0 = 0 2


I


95












NMN ) by using


Mis a hyperfield N



r(n + MN) = (r + M)(n + MN)











MN







MN and so vdimFN

MN = p





96



dimR le vdim RM

MM2


M




M

MM2
















Pe0 sub Pe

1 sub sub Pen




97



Qc0 sub Qc

1 sub sub Qcn








PRP = (n

sumi=1

Rai)RP =n

sumi=1

RPai1




5 Conclusions


M2








98








99

mathematics

Article









MSC 20N25 20N20

1 Introduction



101





2 Preliminaries


21 Fuzzy Multisets






M = a1n1 aknk

andM = a1 a1︸︷︷︸

n1

ak ak︸︷︷︸nk

102



databases [8]


Projgrade = 60 70 85 90 95


M = 60 60 70 85 85 90 90 95 = 602 701 852 902 951





micro1A(x) micro2



pA(x)


A(x)


CMA(0) = CMA(2) = 0 CMA(1) = (05 05 03 01 01) CMA(3) = (065 02 02 02)





A = (04 02)0 (05 03 01)1 (065 02 02 02)3

B = (05 03 01 01)1 (05 01)2 (065 02 02)3


A = (04 02)0 (05 03 01 0)1 (0 0)2 (065 02 02 02)3

B = (0 0)0 (05 03 01 01)1 (05 01)2 (065 02 02 0)3

103






jA(x) = micro



jAcapB(x) = micro

jA(x) and micro



jA(x) or micro



A = (05 03 01 01)1 (065 02 02 02)3

B = (05 03 01 01)1 (05 01)2 (065 02 02 02)3

C = (05 03)0 (05 01)2 (035 03 01)3





CM f (A)(y) =


0 otherwise


22 Hypergroups



A B =⋃

aisinAbisinB






104


is a hypergroup










x y =


2Z+ 1 otherwise











105





a b

a H a

b a b



1 0 1 2

0 0 1 0 1 H1

1 0 1 0 1 H1

2 H1 H1 2



CMA(x) =





Proof






106







(micro1A(x) micro2


A(x) gt 0









(micro1A(x) micro2



107












CM(x) =

t if x isin S

0 otherwise


CMα =

H if α = 0

S if 0 lt α le t





108






A = (02 01)a (05 04 04)b B = (07 005 005)a (07 005 005)b





2


A = (08 04 04)a (01 01)b B = (06 04 04 04)a (03 01 01)b

Then A⊎

B = (07 04 04 02)a (02 01 005)b






2 We












A(x) micropA(x))

109









CM(x) =

CMA(x) if x isin S

0 otherwise




CM(x) =

CMA(x) if x isin S

0 otherwise



CM(x) =

0 if x isin S

CMA(x) otherwise=


CMA(x) otherwise


110





















orf (x)=y1


111


orf (x)=y2


orf (x)=y CMA(x)

and CMA(a) =or











5 Conclusions






112


References





















113



12 168 [CrossRef]


114

mathematics

Article










Massouros







nal affiliations








40)






1 Introduction




115












y(x) = 0


yprime1 = y2








sumk=0

pk(x)y(k)(x) (2)



116




sumk=0

pky(k) (4)




















117




L(qn q0)y(x) =n

sumk=0




22 Results








LA0(J)id01 LA1(J)

id12 LA2(J)id23

LA0(J)

id0

OO

LA1(J)

id1

OO

F1oo

φ1

ii

LA2(J)

id2

OO

F2oo

φ2hh

F3oo

φ3gg



LAnminus2(J)

idnminus2

OO

LAnminus1(J)

idnminus1

OO

Fnminus1oo

φnminus2ii

LAn(J)

idn

OO

Fnoo

φnhh

Fn+1oo

φn+1gg

(11)



118























119




δ(δ(a r s)) = δ(a r s) (12)







32 Results



dxn +nminus1

sumk=0

pk(x)dk f (x)

dxk (14)



where


dxn +nminus1

sumk=0


dxk (16)



by



sumk=0


120





isin LA1An(R) Then


= δn

(m


nminus1

sumk=0


)=

= δn


nminus1

sumk=0


)=

= kdn f (x)


sumk=0

pk(x) +nminus1

sumk=0

qk(x) + k =


sumk=0

(pk(x) + qk(x)) =


sumk=0

(pk(x) + qk(x)) =

= (m + k)dn f (x)


sumk=0

(pk(x) + qk(x)) =







by






[L(m


minusminusrarrq(x))

)le

=[


minusminusrarrq(x)

)le

=






121
























= τn

(L(m pnminus1 p0)

n

sumk=0

(ak + bk)x

)=



122







ψn( f ) = ψn

(n

sumk=0

akxk

)=

nminus1

sumk=0







φntimesψn

LA1An(R)

φn



(32)




(τn

(L(m pnminus1 p0)

n

sumk=0

akxk

))=


= τnminus1

(L(m pnminus2 p0)

nminus1

sumk=0

akxk

)=

= τnminus1

((φn times ψn)

(L(m pnminus1 p0)

n

sumk=0

akxk

))=









123




dx2 + p1(x)dy(x)

dx+ p0(x)y(x) (35)






k=0 bkxk Then


= ψn( f )ψn(g)





Rr[x] then


(r

sumk=0

akxk

)= rednminus1

(r

sumk=n

akxk +nminus1

sumk=0

akxkminus1

)=

=nminus1

sumk=0




dxn +nminus2

sumk=0

pk(x)dky(x)

dxk (38)



dxnminus1 +nminus2

sumk=0

pk(x)dky(x)

dxk (39)


we obtain


124


and


n

(L(m pnminus1 p0)

nminus1

sumk=0

akxk

)=




sumnk=0 bkxk Then




= λn

(L(m pnminus1 p0)

n

sumk=0

(ak + bk)xk

)=





LA1Anminus1(R)

ηnminus1

LA1An(R)timesRn[x]

λn LA1An(R)

(43)


mdnminus1





and ηnminus1

(sumnminus1


k=0 akxk


above we have










125



yr(t) =n

sumk=1

wrk(t)xrk(t) + br









k=1wk(t)xk(t) +



ys(t) =n

sumk=1

wsk(t)xsk(t) + bs











126



y(t) =n

sumk=1

wk(t)dkminus1 p(t)


dtn (47)


y(t) =n

sumk=1

wk(t)dkminus1 p(t)


dtn

and

z(t) =n

sumk=1

wk(t)dkminus1q(t)


dtn


u(t) = y(t) + z(t) =n

sumk=1



dtn








nsum








p






127





ys(t) =n

sumk=1k 6=m


r


((w2rm(t) + 1)wr1(t) w3










0 = e1 = wrm(t)ws1(t) + wr1(t)

0 = e2 = wrm(t)ws2(t) + wr2(t)





wsm(t) =1


wrn(t)wrm(t)



wrm(t)

1wrm(t)

minus wrn(t)wrm(t)

)=

=1


128




5 Conclusions























129






130

mathematics

Article








1 Introduction




131


2 Preliminaries












132





1 2 31 1 1 12 1 2 33 1 2 3 3 2 3



1 2 3 4 51 1 2 3 4 52 2 1 3 4 53 3 3 123 5 454 4 4 5 124 355 5 5 45 35 12345










133





1 2 3 4 5 61 1 2 3 4 5 62 2 1 4 3 6 53 1 2 3 4 5 64 2 1 4 3 6 55 135 246 135 246 135 2466 246 135 246 125 246 135












1 2 3 41 1 2 3 42 2 1 2 4 3 43 3 4 1 3 2 44 4 3 4 2 4 1 2 3 4

134




1 2 3 4 5 61 1 2 3 4 5 62 2 1 3 5 4 63 3 3 1 2 6 6 4 54 4 5 6 1 2 35 5 4 6 2 1 36 6 6 4 5 3 3 1 2





Proof





as required





135



1 2 3 41 1 1 2 1 12 1 1 2 1 13 1 2 3 4 34 1 2 3 4 4




Proof






1 2 3 4 5 61 1 2 3 4 5 62 2 1 3 5 4 63 3 3 12 6 6 454 4 5 6 1 2 35 5 4 6 2 1 36 6 6 45 3 3 12






136





4 Conclusions





References






47 2209ndash2219


137

mathematics

Article







math9020108







nal affiliations








40)






1 Introduction





139






⋃xisinAyisinB x y











140










3 Main Results




⋃

xisinAj

ϕ(a x) =⋃

xisinAi

ϕ(x b) = Aij (1)


a Y =⋃

yisinYa y X b =

⋃

xisinX= x b X Y =

⋃

xisinXyisinYx y



all x y isin H let

x y =



141












(x y) z = (x y) z =⋃

aisinxya z = A(rs)t



x H =⋃

yisinHx y =

( ⋃

yisinAj

x y)⋃( ⋃

yisinHminusAj

x y)

=(

x Aj

)⋃( ⋃

risinGminusjAir



yisinH x y =⋃






142





a b

c c dd d c


a b c d



(c a) a = c sub A2 c (a a) = c A1 = A2










143










1 2 3

1 1 2 3 2 32 2 3 1 13 2 3 1 1

1 2 3 4

1 1 2 3 4 2 3 4 2 3 42 2 3 4 1 1 13 2 3 4 1 1 14 2 3 4 1 1 1

1 2 3 4

1 1 2 3 2 3 42 2 3 4 4 13 2 3 4 4 14 4 1 1 2 3





144










1 1 2 3 4 5

1 1 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 52 2 3 4 5 1 1 1 13 2 3 4 5 1 1 1 14 2 3 4 5 1 1 1 15 2 3 4 5 1 1 1 1


2 1 2 3 4 5

1 1 2 3 4 2 3 4 2 3 4 52 2 3 4 5 5 5 13 2 3 4 5 5 5 14 2 3 4 5 5 5 15 5 1 1 1 2 3 4

145



1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 1 13 2 3 1 14 4 5 1 15 4 5 1 1

By Proposition 5






P 2 3

23

Q 4 5

45



3 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 5 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3


P 2 3

2 4 4 53 4 5 R

Q 4 5

4 2 3 2 35 2 3 S


4 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 3

5 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

146


6 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

7 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 4 5 1 13 2 3 4 5 4 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3


P 2 3

2 43

Q 4 5

4 2 3 2 35 2 3 2 3


8 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

9 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 5 4 1 13 2 3 4 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

10 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 5 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

11 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 5 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

12 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

13 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 4 5 4 1 13 2 3 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

14 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 5 4 1 13 2 3 4 5 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3

15 1 2 3 4 5

1 1 2 3 2 3 4 5 4 52 2 3 5 4 1 13 2 3 4 5 1 14 4 5 1 1 2 3 2 35 4 5 1 1 2 3 2 3


147



16 1 2 3 4 5

1 1 2 3 2 3 4 52 2 3 4 4 5 13 2 3 4 4 5 14 4 5 5 1 2 35 5 1 1 2 3 4


17 1 2 3 4 5

1 1 2 3 2 3 4 52 2 3 1 1 5 43 2 3 1 1 5 44 4 5 5 2 3 15 5 4 4 1 2 3


18 1 2 3 4 5

1 1 2 3 2 3 4 52 2 3 1 1 5 43 2 3 1 1 5 44 4 5 5 1 2 35 5 4 4 2 3 1








148




1 2 3 4 5 6

1 1 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 2 3 4 5 62 2 3 4 5 6 1 1 1 1 13 2 3 4 5 6 1 1 1 1 14 2 3 4 5 6 1 1 1 1 15 2 3 4 5 6 1 1 1 1 16 2 3 4 5 6 1 1 1 1 1


1 2 3 4 5 6

1 1 2 3 4 5 2 3 4 5 2 3 4 5 2 3 4 5 62 2 3 4 5 6 6 6 6 13 2 3 4 5 6 6 6 6 14 2 3 4 5 6 6 6 6 15 2 3 4 5 6 6 6 6 16 6 1 1 1 1 2 3 4 5


1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 62 2 3 4 5 5 5 6 13 2 3 4 5 5 5 6 14 2 3 4 5 5 5 6 15 5 6 6 6 1 2 3 46 6 1 1 1 2 3 4 5


1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 62 2 3 4 1 1 1 6 53 2 3 4 1 1 1 6 54 2 3 4 1 1 1 6 55 5 6 6 6 2 3 4 16 6 5 5 5 1 2 3 4

149






1 2 3 4 5 6

1 1 2 3 2 3 4 5 4 5 62 2 3 6 6 1 1 4 53 2 3 6 6 1 1 4 54 4 5 1 1 6 6 2 35 4 5 1 1 6 6 2 36 6 4 5 4 5 2 3 2 3 1


1 2 3 4 5 6

1 1 2 3 2 3 4 5 4 5 62 2 3 X Y 6 6 13 2 3 Z T 6 6 14 4 5 6 6 1 1 2 35 4 5 6 6 1 1 2 36 6 1 1 2 3 2 3 4 5




150



1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 62 2 3 4 1 1 1 6 53 2 3 4 1 1 1 6 54 2 3 4 1 1 1 6 55 5 6 6 6 1 2 3 46 6 5 5 5 2 3 4 1


1 2 3 4 5 6

1 1 2 3 2 3 4 5 4 5 62 2 3 1 1 6 6 4 53 2 3 1 1 6 6 4 54 4 5 6 6 1 1 2 35 4 5 6 6 1 1 2 36 6 4 5 4 5 2 3 2 3 1


1 2 3 4 5 6

1 1 2 3 2 3 4 5 62 2 3 4 4 5 6 13 2 3 4 4 5 6 14 4 5 5 6 1 2 35 5 6 6 1 2 3 46 6 1 1 2 3 4 5




1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 6 5 62 2 3 4 5 6 5 6 5 6 1 13 2 3 4 5 6 5 6 5 6 1 14 2 3 4 5 6 5 6 5 6 1 15 5 6 1 1 1 2 3 4 2 3 46 5 6 1 1 1 2 3 4 2 3 4


151



1 2 3 4 5 6

1 1 2 3 4 2 3 4 2 3 4 5 6 5 62 2 3 4 1 13 2 3 4 1 14 2 3 4 1 15 5 6 1 1 16 5 6 1 1 1


P

2 3 4

234

Q 5 6

56




Q 5 6

5 2 R6 S T



Q 5 6

5 R 26 T S




Q 5 6

5 2 3 2 3 46 2 3 4


152



Q 5 6

5 R 2 36 S T




P

2 3 4

2 5 5 6 5 63 5 6 R 5 64 5 6 5 6 S

Q 5 6

5 2 3 4 2 3 46 2 3 4 T



P

2 3 4

2 534

Q 5 6

5 2 3 4 2 3 46 2 3 4 R







Hypergroups 13 13 3 12 12 1180 1233


153




Case 1 2 3 4 5 6 7 8 9 Total












154



















155

mathematics

Article







1 Introduction

11 Motivation


157















158



















x y =





159


















Ancc =

((n⋃

i=1

Ii )

notimes

i=1

Sin

prodi=1

δi Mnn(ϕ)

)

wherenprodi=1

δi n⋃

i=1Ii times

notimesi=1

Si rarrnotimes



160


Mnn(ϕ) =

ϕ11 ϕ12 ϕ1nϕ21 ϕ22 ϕ2n

ϕn1 ϕn2 ϕnn

is defined by

(n

prodi=1

δi


=




(7)

and n⋃

i=1Ii times

n⋃i=1

Ii rarr Plowast(

n⋃i=1

Ii


n⋃i=1

Ii defined by

x y =




n

prodi=1

δi

)(y

(n

prodi=1

δi

)(x (s1 s2 sn))

)isin(

n

prodi=1

δi


notimes

i=1

δi(Ii si) (8)


x isinn⋃







161



S

S

S

S

3

2

1

n

Ii

i=1

I

I

I

I

1

2

3

n

Input forSMAIL


A

A

AA

1

23

4







SpRM =

[s A] |s isin Rp

6 A isin Mmn

(9)

where

Rp6 =

s = (s1 s2 s6)|s1 = p s2 s3 s5 isin R+


4

π

4〉

(10)

162


and

Mmn =

a11 a1n

am1 amn

|aij isin

0 for i = m+1

2 j = n+12


(11)







A =


isin Mmn (12)




IpVM =

[~i Cmn

]=

(i1 i2 i6)

c11 c1n

cm1 cmn


langminusπ

4

π

4

rang


(13)

163


AA

AA

1

2

3

4

123456789

1011





VM is q ie

IpVM cap Iq



VM times IpVM rarr

Plowast(IpVM) by





(2 62 0 1 4 minusπ8 )


]]and [~c C23] =

[(2 81 1 2 5 π

6 )


]] Then the

hyperoperarion


[(2 62 0 1 4 minusπ

8)


]]lowastp

[(2 81 1 2 5

π

6)


]]=


8le r6 le

π

6



]


]

164





)=(





⋃


⋃



=



=


=

⋃


⋃



lowastp ~w =

(~u lowastp ~v

)lowastp ~w





)lowastp C



VM =⋃

[~uCmn ]isinIpVM






[~aı ı

mn

]A is


mn]

is a state





δp

([~aı ı

mn

]C[vs Ks

mn])

= (15)

(radic


c11k11 c1nk1n

cm1km1 cmnkmn

s


165


The left hand side

δp

([~aı ı

mn

]A δp

([~bı Bı

mn

][us Us

mn]))

=

δp

([~aı ı

mn

]A[(radic


)s

(bijuij)s])

=

δp

([~aı ı

mn

]A[(radic


)

s (bijuij)

s])

=

[(radic



)

s (aijbijuij)

s]=


s] =[vs Vs

mn]

The right hand side

δp

([~aı ı

mn

]A lowastp

[~bı Bı

mn

][us Us

mn])cup δp

(IpVM

[us Us

mn])


δp

(IpVM

[us Us

mn])

=⋃

[~pı Pımn]isinIp

VM

δp

([~pı Pı

mn

][us Us

mn])


mn

]isin Ip


(p a2i2s+a2i2+a

s+1 a3b3 a4 + b4 a5b5 a6


a11b11 a1nb1n

am1bm1 amnbmn

i

Thus

[vs Vs

mn]= δp

([~cı Cı

mn

][us Us

mn])isin

⋃

[~pı Pımn]isinIp

VM

δp

([~pı Pı

mn

][us Us

mn])





[as Bs

mn]isin S3



mn


166



mn =



then the


[rs Rs




[rs Rs

mn]



-1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 -1

-1 -1 -1 1 1 1 -1 -1 -1

-1 -1 -1 1 1 1 -1 -1 -1

-1 -1 -1 1 1 1 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 1 1 1 -1 -1 -1

-1 -1 1 1 1 -1 -1 -1

-1 -1 1 1 1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 -1 1

-1 -1 -1 -1 -1 -1 -1 1 1

-1 -1 -1 -1 -1 -1 -1 1 1

-1 -1 -1 -1 -1 -1 -1 1 1

-1 1 1 1 -1 -1 -1

-1 1 1 1 -1 -1 -1

-1 1 1 1 -1 -1 -1

-1 -1 -1 -1 -1 -1 -1 1 1-1 -1 -1 -1 -1 -1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1-1-1

-1 -1 -1

-1 -1 -1 -1 -1 -1

1 1 1

1 1 11 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1 1

1 1

0 0 0

0 0 0

-1 -1 -1 0 0 0 -1 -1 -1 -1 -1 -1 0 0 0 -1 -1 -1

-1 -1 -1 0 0 0 -1 -1 1

-1 -1 -1 0 0 0 -1 -1 1

-1 -1 -1 0 0 0 -1 1 1

-1 -1 -1 0 0 0 -1 1 1

1

-1-1

-1-1

-1 -1

Lane 3 LaLane 2ne 4





mn

]for

which holds

δp

([~eı Eı

mn

][ss Ss

mn])

=[ss Ss

mn]

(16)

167




words[~eı Eı

mn

]=

(e1 e2 e3 e4 e5 e6)

111 11n 1m1 1mn


s2+1 e3 = 1 e4 =


δp

([~eı Eı

mn

][ss Ss

mn])

=

δp

(p

s2

s2 + 1 1 0 1 s6)i

111 11n 1m1 1mn

i

(s1 s2 s3 s4 s5 s6)s

a11 a1n am1 amn

s

=

(radic

p2s2

s2 + 1s2 +

s2


)

111a11 11na1n

1m1am1 1mnamn

s =

(

ps2(s2 + 1)

s2 + 1s3 s4 s5 s6

)

a11 a1n am1 amn

s =

[ss Ss

mn]



6 aij = 0



RM δp)




pss pSsmn]

is equal to[(

pss1 pss

6)

pbsij

]isin SpRM

and every input[

p~aıpAımn

]is equal to

[(paı

1 paı6)

paıij


MAp = (IpVMSp





ϕpp([(pss

1pss2pss

3pss4pss

5pss6)pAs

mn])

=[(pvı

1pvı2pvı

3pvı4pvı

5pvı6)pBı

mn]

where pvı1 = pss

1 = p pvı2 = pss

2

pss2+

1pss

5

pvı3 = 0 95 pvı

4 = pss4 + l + sgn

(ss

6)

pvı5 = 1

pss5 pvı

6 = 0 and


168







MAp = ((IpVMSp


RM δq)) by

ϕpq([(pss

1 pss2 pss

3 pss4 pss

5 pss6)pAs

nm])

=[(qvı

1 qvı2 qvı

3 qvı4 qvı

5 qvı6)qBı

nm]

(17)

where

qvı1 = q 6= pss

1 qvı2 =

pss2

pss2 + 1


6|10

qvı3 = 0 99 qvı

4 = 0q vı5 = 1q vı

6 = 0


λ =

1 if pss4 gtq ss

4 and pss6 gt 0

1 if pss4 ltq ss

4 and pss6 lt 0

0 otherewise

and α =

1 if pss4 =q ss

4 and qss3 ltp ss

4 and pss5 gt 1

0 otherewise







SMAil =

(( p⋃

i=1

IiVM

)

potimes

i=1

SiRM

p

prodi=1

δi Mnn(ϕ)

)



p

prodi=1

δi

)([~aı Aı

mn

]

(p

prodi=1

δi

)([~bı Bı

mn

]

potimes

i=1

[iss iS

smn]))isin

(p

prodi=1

δi

)([~aı Aı

mn

]lowastp

[~bı Bı

mn

]

potimes

i=1

[iss iS

smn])cup

potimes

i=1δi

(IiVM

[iss iS

smn])

(18)


(t

prodi=1

δi

)([~aı Aı

mn

]

(t

prodi=1

δi

)([~bı Bı

mn

]

totimes

i=1

[iss iSs

mn]))isin

totimes

i=1

δi

(IiVM

[iss iSs

mn])

(19)


169



p~aıpAımn

][

p~bıpBımn

]isin Ip



p~bıpBımn


([1ss 1Ss

mn]

[

2ss 2Ssmn]

[

tss tSsmn])



p~aıpAımn


[p~cıpCı

mn



δp([

p~aıpAımn

] δp(

[p~bıpBı

mn

][

pss pSsmn])) = δp(

[p~cıpCı

mn

][

pss pSsmn])

When the inputs[

p~aıpAımn

]or[

p~bıpBımn


[tss tSs

mn]



p~aıpAımn

]isin Ip

VM[

q~bıqBımn

]isin Iq


1ss 1Ssmn]

[

2ss 2Ssmn]

[

tss tSsmn])




RM δ1)MA2 =

(I2VMS2





([1rs 1Rs

mn]

[

2ss 2Ssmn]

[

3ts 3Tsmn]

[

4us 4Usmn]

[

5vs 5Vsmn])


0 0 0 0 00 ϕ ϕ ϕ 00 ϕ ϕ ϕ 00 ϕ ϕ ϕ 00 0 0 0 0



[(33ı 31ı 3098ı 30ı 31ı 3

π5

ı) 3(bij)ı= 1


170



δ3

([(33ı 31ı 3099ı 30ı 31ı 3

π

5

ı) 3(bij)ı= 1

][

3ts 3Tsmn])

=[(

33 330 319206 33 31 3π

5

)3Ts

nm

](20)


ϕ33

([(33s 330s 319206s 33s 31s 3

π

5

s)3Ts

nm

])=[(33ı 31ı 31ı 31ı 31ı 30s) 3Cı

nm]

where 3Cınm =





[(33 330 319206 33 31 3

π5)

3Tsnm] we obtain

δ3([(33ı 31ı 31ı 31ı 31ı 30ı) 3Cs

nm]

[

3ts 3Tsmn])

=[(33s 330s 319206s 34s 31s 30s) 3Ds

nm]


33s 330s 319206s 33s 31s 3π5

s) 3Tsnm]



ϕ32

([(33s 330s 319206s 33s 31s 3

π

5

s)3Ts

nm

])=

[(22ı 2

(3031

+ 00628)ı

2099ı 20ı 21ı 20ı)

2Mınm

] (21)




[(33s 330s 319206s 33s 31s 3

π5

s) 3Tsnm]


Thus the input[(

44ı 3( 30

30+1)ı

3099ı 30ı 31ı 30s)

3Nımn




171





4 Conclusions


172




References





















173








174

mathematics

Article











MSC 20N20 14H52 11G05

1 Introduction


175




2 Preliminaries


21 Conics


∆ =




Q0(x y) =




176


O

xi bull11

xj

xi

xj

xi bull11

xj

QR ( f11)




)





p3

Conicp6

p4

p5

p2

p1




177



⋃aisinAbisinB




⋃















178








3 Hyperconic






αF =

0 if char(F) = 2

1 if char(F) 6= 2




x times y


where

179













aisinAbisinB

Q( fab F)




Q( f10








O =


Gx( fab) =






O if x 6= 0

0 if x = 0and

x +O =

O if O = 0minus1



180




QF( f AB) =⋃

aisinAbisinB

QF( fab)



and

Lab(xi xj) =







O if x 6= 0

0 if x = 0xoplusO =

O if O = 0minus1





x fab(x) = ax2 + bx





F)


) x1 6= x2 and i j = 1 2 Then



x21x2 minus x2

2x1

b1 = b2 =minusx2

2y1 + x21y2

x21x2 minus x2

2x1




2


2 y1 + xminus11 y2


2


F) as we expected

181






xi bullab xj =



xi minusfab(xi)

f primeab(xi)

xi = xj 6isin O

xi xi 6= O = xj

xj xj 6= O = xi

O (xi xj) = (OO)





0= infin








the conic QF( fab)


F = 0 1 2 QF( f10) = QF( f10) cupQF( f



=


(minus1)0) = O





Theorem 3(QF( f AB)

)is a hypergroup


182










=



Similarly












So


)=





=⋃




183


Otherwise


)









On the other hand








On the other hand


)




x QF( f AB) =(

x ⋃



=( ⋃



=(⋃

iisinAjisinB


= QF( f AB)


)is

a hypergroup


184



10) = QF( f10) cup QF( f







)




)








⋃(st)(ij)isinI


(ij)isinIQF( fij)










185


















( f11) )

O (1 2) (2 1) (2 2)O O (1 2) (2 2) (2 1) (2 2) (1 2)

(1 2) (1 2) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2)(2 1) (2 1) (1 2) (2 1) (2 2) QF( f11) (1 2) (2 1) (2 2)(2 2) (1 2) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2) (1 2) (2 1) (2 2)


(QF( fmn)

)of

(QF( fmn)






186


4 Conclusions



(QF( fnm)

)

is investigated

HYPERCONIC fAB





References






187
























188








189

mathematics

Article
















iations








40)






1 Introduction




andAperpB = cup


191













xperpH = Hperpx = H







192


















193













1 2 3

1 1 2 3

2 3 1 2

3 2 3 1


1 2 3

1 1 3 2

2 2 1 3

3 3 2 1









1 2 3

1 1 2 3

2 3 1 2

3 2 3 1


1 2 3

1 1 3 2

2 2 1 3

3 3 2 1






1 2 3

1 1 1 123

2 1 1 23

3 23 23 123


1 2 3

1 2 3 13

2 12 13 123

3 13 123 123









194






1 2 3

1 1 1 123

2 1 1 23

3 23 23 123


1 2 3

1 2 3 13

2 12 13 123

3 13 123 123



















1 2 3

1 1 12 13

2 13 123 3

3 12 2 123


1 2 3

1 1 13 12

2 12 123 2

3 13 3 123










195





1 2 3

1 1 12 13

2 13 123 3

3 12 2 123


1 2 3

1 1 13 12

2 12 123 2

3 13 3 123






















196



a(bc)= a(

bcminus1)= a

(bcminus1

)minus1= a

(cbminus1


(abminus1

)=

= c(

baminus1)minus1

= cbaminus1 = c(ba)


aH = Ha = H















( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )

11 1 1 1 1 1

11 1


c ba c ba c b a


minusminus minus

= = = = = = =

= = =


a H H a H= =








1 2 3

1 123 123 123

2 13 12 13

3 23 23 1



1 2 3

1 12 123 13

2 12 13 123

3 123 12 13























197


















risinab(rc cup Irc)

]cup[cup


]=

= (ab)c cup[cup

risinabIrc

]cup[cup


]

and


[cup

risincb(ra cup Ira)

]cup[cup


]=

= (cb)a cup[cup

risincbIra

]cup[cup


]





198








( ) ( ) ( ) ( ) ( )

( ) ( )

ab

ab

ab ab rc scr ab s I

rc scr ab s I


ab c I sc I

isin isin

isin isin


= cup

and

( ) ( ) ( ) ( ) ( )

( ) ( )

cb

cb

cb cb ra sar cb s I

ra sar cb s I


cb a I sa I

isin isin

isin isin


= cup


Proposition 4 If

aba b H

Iisin







1 2 3

1 23 123 13

2 12 13 123

3 13 23 12



1 2 3

1 13 12 123

2 123 13 12

3 13 12 23





Proposition 7


aba b H

Iisin



ab

a b HI

isinne empty then












Proposition 7






199











1 2 3

1 a11 a12 a13

2 a21 a22 a23

3 a31 a32 a33





1 2 3

1 1 12 123

2 12 3 13

3 123 123 123




200






1 2 3

1 a11 a12 a13

2 a21 a22 a23

3 a31 a32 a33





1 2 3

1 1 12 123

2 12 3 13

3 123 123 123





1 2 3

1 1 1 123

2 1 1 23

3 123 123 123










201



TotalNumber

IsomorphismClasses


(Rigid)

Classeswith 2

Members

Classeswith 3

Members

Classes with4 or 5

Members

Classeswith 6









16 044 2733 0 6 110 0 2617



4628 800 0 1 56 0 723



0



0




2520 466 6 3 78 0 399

hypergroups 23 192 3999 6 10 244 0 3739




hypergroups 23 192 3 999 6 10 244 0 3 739



1 2 3

1 123 23 23

2 13 123 13

3 12 12 123


1 2 3

1 123 13 12

2 23 123 12

3 23 13 123



H if x y



H if x y




202




hypergroups 23 192 3 999 6 10 244 0 3 739



1 2 3

1 123 23 23

2 13 123 13

3 12 12 123


1 2 3

1 123 13 12

2 23 123 12

3 23 13 123



H if x y



H if x y






xy =



xy =











which is absurd

203

















204












Proposition 12








Corollary 8


(AB)C = (BC)A


C(BA) = A(CB)

Proposition 13


b(ac) = c(ab)

205



(ba)c = (ca)b






Corollary 9


B(AC) = C(AB)


(BA)C = (CA)B






( ) ( )= b a c c a b


( ) ( )= b a c c a b








Corollary 9


( ) ( )= B A C C A B


( ) ( )= B A C C A B





1 2 3

1 1 2 3

2 3 123 13

3 2 12 123


206




1 2 3

1 1 3 2

2 2 123 13

3 3 12 123





























207












Hence x y isin xy









208





















209




















1 2 3 4 5 6 7 1 12 12 1234 1234 12567 12567 12567 2 1 12 1234 1234 12567 12567 12567 3 1234 1234 34 4 34567 34567 34567 4 1234 1234 3 34 34567 34567 34567 5 12567 12567 34567 34567 57 7 67 6 12567 12567 34567 34567 67 7 7 7 12567 12567 34567 34567 67 67 67













210




Proposition 28




Proposition 29








Proposition 30






211





Proposition 31






Proposition 32










212






KK = KK = K



Proposition 35














213



(xminus1)minus1


Proposition 38



xminus1 e









ex = exminus1 =

e xminus1= xe




e xminus1


e xminus1







214








sube ev e v













1 2 3

1 1 12 13

2 13 123 13

3 12 12 123

















Proposition 47




215


















216


Appendix A














217









End[]EndPackage[]


1 1 231 1 23123 23 13


In[1]=LRHtest[1 1 2 3 1 1 2 3 1 2 3 23 1 3]

And the output is

Out[1]=2







[CrossRef]

218




















219





















220







221



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