m-solid varieties of algebras

M-SOLID VARIETIES OF ALGEBRAS

Advances in Mathematics

VOLUME 10

Series Editor:

J. Szep, Budapest University of Economics, Hungavy

Advisory Board:

S-N. Chow, Georgia Institute of Technology, U S A .

G. Erjaee, Shiraz University, Iran

W. Fouche, University of South Africa, South Africa

P. Grillet, Tulane University, U.S.A

H.J. Hoehnlte, Institute of Pure Mathematics of the Academy of Sciences, Germany

F. Szidarovszky, University ofAirzona, U.S.A.

P.G. Trotter, University of Tasmania, Australia

P. Zecca, Universitd di Firenze, Italy

M-SOLID VARIETIES OF ALGEBRAS

J. KOPPITZ Universitat Potsdam, Germany

K. DENECKE Universitat Potsdam, Germany

Q - Springer

Libraryof Congress Control Number: 2005936714

Printed on acid-free paper.

AMS Subject Classifications: 08605, 08B15, 20M07, 16Y60

0 2006 Springer Science+Business Media, Inc.

All rights rcscrvcd. This work may not bc translatcd or copicd in wholc or in part without thc writtcn permission of the publisher (Springer Science+Business Media, Tnc., 233 Spring Street, New York, NY 10013, USA), cxccpt for bricf cxccrpts in conncction with rcvicws or scholarly analysis. Usc in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not idcntificd as such, is not to bc takcn as an cxprcssion of opinion as to whcthcr or not thcy arc subjcct to proprietary rights.

Printcd in thc Unitcd Statcs of Amcrica.

Contents

Preface vii

1 Basic Concepts 1 1.1 Subalgebras and Homomorphic Images . . . . . . . 1 1.2 Direct and Subdirect Products . . . . . . . . . . . . 11 1.3 Term Algebras, Identities. Free Algebras . . . . . . 16 1.4 The Galois Connection ( Id . Mod) . . . . . . . . . . 24

2 Closure Operators and Lattices 29 2.1 Closure Operators and Kernel Operators . . . . . . 29 2.2 Complete Sublattices of a Complete Lattice . . . . 31 2.3 Galois Connections and Complete Lattices . . . . . 34 2.4 Galois Closed Subrelations . . . . . . . . . . . . . . 37 2.5 Conjugate Pairs of Additive Closure Operators . . . 41

3 M-Hyperidentities and M-solid Varieties 49 3.1 M-Hyperidentities . . . . . . . . . . . . . . . . . . . 49 3.2 The Closure Operators X&, Xg . . . . . . . . . . . 62 3.3 M-Solid Varieties and their Characterization . . . . 64 3.4 Subvariety Lattices and Monoids of Hypersubstitutions 67 3.5 Derivation of M-Hyperidentities . . . . . . . . . . . 71

4 Hyperidentities and Clone Identities 77 4.1 Menger Algebras of Rank n . . . . . . . . . . . . . 78 4.2 The Clone of a Variety . . . . . . . . . . . . . . . . 84

5 Solid Varieties of Arbitrary Type 87 5.1 Rectangular Algebras . . . . . . . . . . . . . . . . . 87

. . . . . . . . . . . . . . . . . . . . . . 5.2 Solid Chains 96

vi Contents

6 Monoids of Hypersubstitutions 103 6.1 Basic Definitions . . . . . . . . . . . . . . . . . . . 103 6.2 Injective and Bijective Hypersubstitutions . . . . . 109 6.3 Finite Monoids of Hypersubstitutions of Type ( 2 ) . 114 6.4 The Monoid of all Hypersubstitutions of Type ( 2 ) . 119 6.5 Green's Relations on H y p ( 2 ) . . . . . . . . . . . . . 126 6.6 Idempotents in H y p ( 2 , 2 ) . . . . . . . . . . . . . . . 138 6.7 The Order of Hypersubstitutions of Type (2 '2 ) . . 151 6.8 Green's Relations in H y p ( n , n) . . . . . . . . . . . 171 6.9 The Monoid of Hypersubstitutions of Type (n) . . . 181 6.10 Left-Seminearrings of Hyper~ubstit~utions . . . . . . 184

7 M-Solid Varieties of Semigroups 197 7.1 Basic Concepts 011 M-Solid Varieties of Semigroups 197 7.2 Regular-solid Varieties of Semigroups . . . . . . . . 206 7.3 Solid Varieties of Semigroups . . . . . . . . . . . . 252 7.4 Pre-solid Varieties of Semigroups . . . . . . . . . . 255 7.5 Locally Finite and Finitely Based M-solid Varieties 260

8 M-solid Varieties of Semirings 267 8.1 Necessary Conditions for Solid Varieties of Semirings 267 8.2 The Minimal Solid Variety of Semirings . . . . . . . 269 8.3 The Greatest Solid Variety of Semirings . . . . . . 271 8.4 The Lattice of all Solid Varieties of Semirings . . . 275 8.5 Generalization of Normalizations . . . . . . . . . . 283 8.6 All Pre-solid Varieties of Semirings . . . . . . . . . 298

Bibliography 321

Glossary 331

Index 337

Preface

The general study of varieties of algebras with finitary operations was initiated by Garrett Birkhoff in the 1930's. He derived the first significant results in this subject and further developments by Al- fred Tarski, and later, for congruence distributive varieties by Bjarni Jbnsson, laid the groundwork for many of the results on lattices of varieties. Varieties are equationally definable classes of algebras of the same finitary type. Identities are used to classify algebras of the same type into varieties. The relation given by "an algebra satisfies an equation as an identity" defines a Galois connection between the class of all algebras of the same type and the set of all equations of this type. This Galois connection (Id, Mod) consists of the operator Id , which associates to each class of algebras, of the given type the set of all identities satisfied in this class and Mod, which maps each set of equations to the class of all algebras satisfying these equations as identities. The connection defines two closure operators IdMod and ModId. The fixed points under the operator ModId are the varieties and the fixed points under IdMod are called equational theories. The collection of all varieties of algebras of a given type forms a complete lattice. Birkhoff's variety theorem characterizes varieties as classes of algebras of the same type which are closed under taking of arbitrary subalgebras, homomorphic images and direct products. The equational theories of a given type form another complete lattice which is dually isomorphic to the lattice of all varieties. Equational theories can be characterized as sets of equations which are closed under arbitrary applications of the five derivation rules for identities. An important area of activity in Universal Algebra has been to try to classify all varieties of algebras or dually all equational theories, of a given type. Even if the type contains only one binary operation symbol, and even if the binary operation symbol satisfies the associative law, so that the algebras are semigroups, such a project is almost hopelessly complicated.

viii Preface

An identity is a formula in a first order language with equality where the variables are bound by the universal quantifiers. The associative law

V X ~ V X ~ V X ~ ( Z ~ (x2 . 23) M (xl . 2 2 ) ~ 3 )

is satisfied (true) in an algebra with a binary fundamental operation if when any elements of the universe are substituted for x l , x2, x3 and the operation symbol is replaced by the operation of the algebra, the resulting elements from the universe are equal.

The fundamental operations +R, e R of a ring R are associative. In the equation

we can replace the binary operation symbol F by either +R or e R

and obtain an identity satisfied in R. This leads us viewing the associative law as a formula of the form

This is an example of a formula in a second order language, where now quantification of operation symbols over the set of all fundamental operations of the ring R is allowed. If substitution of any binary term operation from an algebra A for the binary operation symbol F leads to identities satisfied in A we call the formula (*) a hyperidentity satisfied in A. Any substitution of a binary term for the binary operation symbol F is called a hypersubstitution. Hypersubstitutions can be extended to mappings defined on the set of all terms. The concept of a hypersubstitution plays a crucial role in this book and was defined first in [56]. Like ordinary substitutions, hypersubstitutions can be composed and together with an identity hypersubstitution which maps any operation symbol fi to an ni-ary "fundamental" term fi(xl, . . . , xn,) one obtains a monoid IFtyp(r). Submonoids M of IFtyp(r) can be used to define the weaker concept of an M-hyperidentity. The relation given by "an algebra M-hypersatisfies an equation" defines a second Galois connection between the class of all algebras of the same type and the set of all equations of this type. The operator HMId associates to each class of algebras of the given type the set of all M-hyperidentities satisfied in this class. The operator HMMod maps each set of equations

Preface ix

of type T to the class of all algebras satisfying these equations as M-hyperidentities. The pair (HhIId, HMMod) is a Galois connection between the class of all algebras of a given type and the set of all equations of this type.

The fixed points under the closure operator HMModHMId are called M-solid varieties and the fixed points under HMIdHMMod are called M-hyperequational theories. The collection of all M- solid varieties of a given type forms a complete sublattice of the lattice of all varieties of this type; and if M I is a submonoid of the monoid M Z of hypersubstitutions, then the complete lattice of all M2-solid varieties is a complete sublattice of the lattice of all MI-solid varieties. This leads us to consider a Galois connection between submonoids of IFtyp(r) and sublattices of the lattice of all varieties of algebras of the given type. Instead of the very complex lattice L(r) of all varieties of type T one may consider the lattices Snr(r) of all M-solid varieties of type r . There is some hope that these lattices are easier to describe. Instead of the lattice C(7) of all varieties of type T we will also consider the lattices L(V) of all subvarieties of some given varieties V and the lattices SM(V) of its M-solid subvarieties. Dually, we consider the complete lattice of all M-hyperequational theories which are characterized as sets of equations closed under the five derivation rules for identities and one more rule of consequences, the so-called M-hypersubstitution rule.

Any variety can be described by only one structure, the clone of the variety. Conversely, every clone can be regarded as the clone of some variety. The clone of a variety V is a multi-based algebra where the different sorts are the sets of n-ary terms over the variety V and where the fundamental operations describe the superposition of terms. There is also a one-based variant of this concept which is called a unitary Menger algebra of rank n ([66]). Hypersubsti- tutions correspond to substitutions of the clone of the variety, and identities of the clone of the variety V can be regarded as hyperidentities in the variety V. A variety V is solid if and only if its clone is relatively free. Note that clones are equivalent to algebraic theories, i.e. particular categories, in the sense of I?. W. Lawvere.

x Preface

The aim of this book is to develop the theory of M-solid varieties as a system of mathematical discourse, applicable in several con- crete situations. The general theory will be applied to two classes of algebraic structures, to semigroups and to semirings. Both these varieties and their subvarieties play an important role in Computer Science. The study of rational languages is related to the theory of classes of finite semigroups and monoids. Samuel Eilenberg's correspondence theorem ([42]) connects certain classes of regular languages, so-called varieties of languages, with pseudovarieties of semigroups. Pseudovarieties are classes of finite algebras of the same type which are closed under the formation of homomorphic images, subalgebras and finite direct products. Since the "finite part" of a variety is a pseudovariety (but not conversely), it makes sense to study varieties and M-solid varieties first. For applications of semirings and classes of semirings in Automata Theory and in the theory of Formal Languages see [51]. Type (2) algebras seem to be simple enough to be accessible, yet rich enough to provide an interest- ing hyperidentity structure. This is especially true for semigroups, where a lot is known about the lattice of all semigroup varieties. In both classes the associative law plays an important role. Semirings are the first class with two binary operation symbols where lattices of M-solid varieties for some monoids M of hypersubstitutions were completely determined.

In the first chapter we develop the basic concepts of Universal Alge- bra, especially the equational theory. In this chapter we also define different complexity measures of terms or trees, some important subsets of the set of all terms of a given type and the superposition operation for terms. The influence of term complexity on the Galois connection between identities and algebras is shown in Chapter 8.

A unique feature of this book is the use of Galois connections as a main tool. Galois connections form the abstract framework not only for classical and modern Galois theory, involving groups, fields and rings, but also for many other algebraic, topological, order- theoretical, categorical and logical theories. This concept, with the related topics of closure operators, complete lattices, Galois closed subrelations and conjugate pairs of completely additive closure operators are used throughout the whole book and are introduced in

Preface xi

Chapter 2. Here three different methods are described to characterize complete sublattices of a complete lattice; by using a conjugate pair of completely additive closure operators, by special closure or kernel operators defined on the given complete lattice, and by so- called Galois closed subrelations. The theory of conjugate pairs of completely additive closure operators starts with a relation between two given sets and the Galois connection induced by this relation. The two closure operators obtained from this Galois connection define two complete lattices. A subrelation of the given relation defines a new Galois connection and two more complete lattices, which can also be defined as sets of fixed points of two other closure operators forming a conjugate pair and having the property to be completely additive. It turns out that under some conditions the last two lattices are complete sublattices of the first ones. This theory will be used to characterize lattices of M-solid varieties as complete sublattices of the lattice of all varieties of the given type. The theory of conjugate pairs of completely additive closure operators was used in [16] to characterize lattices of M-solid quasivarieties and in [81] to characterize lattices of M-solid pseudovarieties. A completely different application of this theory is discussed in [96]. Here the basic Galois connection is the connection (Pol , Inv) between operations and relations defined on the same set A and induced by the relation "an operation f preserves a relation Q". The Galois closed sets under the operator Pollnu are clones. A second Galois connection is given by a group G of permutations on A and the relation that a permutation conjugates a clone onto itself. The Galois closed sets on the clone side are the lattices CG of all clones that are closed under conjugation by all members of some permutation group G.

In Chapter 3 we develop the theory of M-hyperidentities and M- solid varieties and in Chapter 4 we show the connection between clone theory and the theory of hyperidentities.

In Chapter 5 we show that there are non-trivial solid varieties of arbitrary type. For each non-trivial type there are infinite chains of solid varieties. This is surprising since the satisfaction as a hyperidentity is quite a strong requirement.

In Chapter 6 we deal with monoid- and semigroup-theoretical prop-

xii Preface

erties of hypersubstitutions. The Galois connection between subvariety lattices and monoids of hypersubstitutions motivates the study of monoid or semigroup properties of the set of all hypersubstitutions and its submonoids. An example is Green's relation C. If two hypersubstitutions are C-related, then their kernels are equal. We will prove that the kernel of a hypersubstitution is an equational theory. Therefore we have two equal equational theories or two equal varieties.

We determine all injective and all bijective hypersubstitutions, study all finite monoids of hypersubstitutions of type (2), determine all idempotent hypersubstitutions of type (2) and of type (2,2), and consider regular elements and the order of hypersubstitutions. To determine the order of all hypersubstitutions of type (2,2) we have to consider several cases, sub- and subsub-cases. This part looks quite technical and includes detailed calculations. The main idea which is used throughout this chapter is the connection between the complexity of a term and the complexity of its image after ap- plying a hypersubstitution. It turns out that the formulas given in [38] are very useful. Moreover, in Chapter 6 we deal with Green's relations on monoids of hypersubstitutions of different types.

In Chapter 7 we present results on M-solid varieties of semigroups for various choices of the monoid M. We determine the least and the greatest elements in the lattices of all regular-solid, pre-solid and solid varieties of semigroups. Moreover, we prove a theorem which characterizes all regular-solid varieties of semigroups. From this result we derive characterization theorems for solid and pre- soild varieties. Using the general theory from Chapter 3 the greatest M-solid variety of semigroups is the M-hyperequational class defined by the associative law, i.e. the class of all algebras of type (2) where the associative law is satisfied as an M-hyperidentity. It is not clear whether such classes are finitely based by identities. This question is discussed in section 7.5. We will also answer the question of which monoids M of hypersubstitutions have the property that the greatest M-solid variety is locally finite. One of the main methods to be used here is the solution of the word problem in a two-generated free algebra over the considered M-solid varieties of semigroups.

Preface xiii

In the last chapter we determine all solid and all pre-solid varieties of semirings. While there are infinitely many solid varieties of semigroups, the lattice of all solid varieties of semirings is a chain consisting of precisely four elements, and there are altogether 13 pre-solid varieties of semirings. These pre-solid varieties are normalizations and 2-normalizations of the solid ones. Calling a variety k-normal when both sides of any of its nontrivial identities have complexity at least k, we extend several central results on normal varieties to the k-th level and apply the results to pre-solid varieties of semirings.

The material of the last three chapters of this book is based on results which are contained in the habilitation thesis of the second author, and in the Ph.D. theses of Dr. Hippolyte Hounnon and Dr. Thawhat Changphas. The authors would like to thank their colleagues in the growing "hyperidentity family" around the world for encouragement and helpful discussions. The support given by the Institute of Mathematics and the Faculty of Sciences of Uni- versity of Potsdam during the summer semester 2004 is gratefully acknowledged.

Prof. Dr. Dr. hc. Klaus Denecke PD Dr. habil. Jorg Koppitz

Potsdam, September 2005

1 Basic Concepts

1.1 Subalgebras and Homomorphic Im- ages

An algebra is a non-empty set A together with a set of finitary operations defined on this set. An n-ary operation on A is for n > 1 a function f A : An + A from the n-th Cartesian power of A into A. We let On(A) be the set of all n-ary operations defined on A and

00

let O(A) := U On(A) be the set of all finitary operations defined n=l

on A.

Remark 1.1.1 1. Any n-ary operation f on A can be regarded as an (n + 1)-ary relation defined on A, called the graph of f . This relation is defined by {(al, . . . , a,+l) € An+' I f (al , . . . , a,) = an+l}.

2. If B c A is a non-empty subset of A, then the restriction f A l B A of f A to B is defined by f A I ~ ( a l , . . . ,a,) := f (a l , . . . ,a,) for all

a l , . . . , an E B.

3. The definition of an n-ary operation can be extended in the following way to the special case that n = 0, for a nullary operation. We define A0 := (8). A nullary operation is defined as a function f : (0) + A. This means that a nullary operation on A is uniquely determined by the element f (0) E A. For every element a E A there is exactly one mapping fa : (0) + A with fa(@) = a. Therefore a nullary operation may be thought of as selecting an element from the set A. If A = 0 then there are no nullary operations on A.

There are two ways to view an algebra, in the non-indexed form as a pair A := (A; FA) consisting of a set A and a set FA of operations defined on A or as an indexed algebra.

Definition 1.1.2 Let A be a non-empty set. Let I be some non- empty index set, and let (ft)i,l be a function which assigns to

2 1 Basic Concepts

every element i of I an ni-ary operation ft defined on A. Then the pair A = (A; (f:)i,I) is called an (indexed) algebra with the base set or carrier set or universe A, and with (f:)iE1 as the sequence of fundamental operations of A. The sequence r := (ni)iE1 of all arities of the fundamental operations ft is called the type of A. We use the name Alg(r) for the class of all algebras of type r .

In our definition we do not allow the base set A of an algebra to be the empty set. It is possible to define an empty algebra with the empty set as universe if no nullary operations belong to the type. But mostly we will exclude this case. In the indexed case the fundamental operations form a sequence, not a set, that means, repititions are possible. We will give some examples for our definition.

An algebra (G; .) of type r = (2), i.e. having one binary operation, is called a groupoid. The binary operation is simply denoted by . but we will also write z y or just z y . If the binary operation is associative, G is called a semigroup. A semigroup with an additional nullary operation e is called a monoid M = (M; 0 , e) if e is an identity element with respect to ., i.e. if for all x E M the equations x . e = e . x = x are satisfied. Monoids are algebras of type r = (2,O).

A group G = (G; a ) is a semigroup, which satisfies the axiom

' d a , b ~ G 3 x , y ~ G ( a . x = b a n d ~ . a = b ) (invertibility) .

A group can also be regarded as an algebra G = (G; ., -I, e) of type (2, 1, 0), where the associative law and the axioms

and 'dx E G ( x - e = x = e . x )

are satisfied.

An algebra R = (R; +, .) of type r = (2,2) is called a semiring if both these binary operations are associative and the distributive laws

'dx, y, Z E R ( x . ( ~ + z ) = x . ~ + x . z )

I. 1 Subalgebras and Homomorphic Images

and

are satisfied.

Rings are semirings with commutative and invertible addition and a commutative ring K = (K ; +, 0 ) is called a field if ( K \ (0); 0 ) is a group, where the zero element 0 is the neutral element with respect to addition.

An algebra V = (V; A, V) of type (2,2) is called a lattice, if the following equations are satisfied by its two binary operations, which are usually called meet and join:

'dx, y E V 'dx, y E V 'dx,y,z E V 'dx, y, x E V 'dx E V 'dx E V 'dx,y E V 'dx,y E V

If in addition the lattice satisfies the following distributive laws, ' d x , ~ , ~ E V (XA(YVZ) = (XAY)V(XAZ)) ,

'dx, y, x E V (xV(yAx) = (zVy)A(xVx)), then the lattice is said to be distributive.

Lattices are important both as examples of a kind of algebra, and also in the study of all other kinds of algebras, since any algebra has some lattices associated with it. A lattice can also be regarded as a partially ordered set (V; I), where I is a binary, reflexive, anti- symmetric and transitive relation on V. There is a close connection between lattices and partially ordered sets, as the following theorem shows.

Theorem 1.1.3 Let (V; 5) be a partially ordered set in which for all x, y E V both the in f imum A{x, y) and the supremum V{x, y) exist. T h e n the in f imum and supremum operations make (V; A, V)

4 1 Basic Concepts

a lattice. Conversely, every lattice defines a partially ordered set i n which for all x, y the in f imum A{x, y } and the supremum V{x, y } exist.

Proof: Let (V; 5) be a partially ordered set, in which for any two- element set {x, y ) & V the infimum A{x, y ) and the supremum V{x, y ) exist. We define x A y := A{x, y ) and x V y := V{x, y ) . Then the required identities are easy to verify.

If conversely (V; A, V) is a lattice, then we define

and see that this gives a partial order relation on V. It is easy to check that A{x, y ) = x A y and V{x, y ) = x V y are satisfied.

A lattice C in which for all sets B & L the infimum B and the supremum V B exist is called a complete lattice. Obviously, any finite lattice is complete.

A bounded lattice (V; A, V , 0 , l ) is an algebra of type (2, 2, 0, 0) which is a lattice, with two additional nullary operations 0 and 1, which satisfy

'dx E V (XAO = 0) and 'dx E V (xv1 = I ) .

An algebra B = (B; A, V, l , O , 1) is called a Boolean algebra, if (B; A, V, 0 , l ) is a bounded distributive lattice with an additional unary operation 1 satisfying

and

'dx E B ( x V 1 x = I ) .

Important examples of Boolean algebras are the two-element Boolean algebra ((0, 1); A, V , 0 , l ) with conjunction and disjunc- tion (meet and join operation) as binary, and with negation as unary operations and the power set algebra (P(A) ; n, U, N, 0, A) of

I. 1 Subalgebras and Homomorphic Images 5

A with intersection, union, complementation, the empty set and A as operations. An algebra S = ( S ; 0 ) of type (2) is called a semilattice, if the operation . is an associative, commutative, and idempotent binary operation on S . This means that a semilattice is a particular kind of semigroup.

For a given algebra B = ( B ; ( f ? ) i E I ) of type r we obtain new algebras using certain algebraic constructions. The first algebraic construction we want to mention, is the formation of subalgebras.

Definition 1.1.4 Let B = ( B ; (f?)i , I) be an algebra of type r . Then an algebra A is called a subalgebra of B, written as A c B, if the following conditions are satisfied:

(i) A = (A; ( f , i ' ) i ,~ ) is an algebra of type r ;

(ii) A B;

(iii) 'v'i E I, the graph of f ) is a subset of the graph of f?.

The subalgebra property can be checked in the following way:

Lemma 1.1.5 (Subalgebra Criterion) Let B = ( B ; ( f ? ) i E I ) be a n algebra of type r and let A C B be a subset of B for which f ) = f? I A for all i E I . T h e n A = (A, ( f ) ) i E I ) i s a subalgebra of B = ( B , ( f ? ) i E I ) i&f A is closed with respect t o all the operations f? for i E I ; that is, i f f?(Ani) C A for all i E I . ( W e say that f? preserves the subset A of B . )

Clearly, the subalgebra relation is transitive, that is, if A C B C C then A c C.

If B = ( B ; ( f ? ) i , ~ ) is an algebra of type r and if {Aj I j E J) is a family of subalgebras of B with the non-empty intersection A := Aj of its universes, then it is easy to see that A is the

j , J

universe of a subalgebra of B which is called the intersection of the family {Aj I j E J), denoted by n A. This allows us to consider

.it J

the subalgebra

( X ) B := n{A I A C B and X C A }

6 1 Basic Concepts

of B generated by a subset X C B of the universe. The set X is called a generating sys tem of this algebra. The process of subalgebra generation satisfies three very important properties, called the closure properties. This makes the subalgebra generation process an example of a closure operator, which we will consider in more detail in chapter 2.

Theorem 1.1.6 Let B be a n algebra. For all subsets X and Y of B, the following closure properties hold:

(i) X C (X)u, ( extensivity) ; (ii) X c Y + (X)u c (Y)u, (monotonic i ty) ; (iii) (X)u = ( (X)u)a , ( idempotency).

If the intersection of two subalgebras of B , is non-empty, then it is again a subalgebra of B , and we would like to use this to define a binary operation on the set S u b ( B ) of all subalgebras of B . Therefore a binary operation A on S u b ( B ) is defined by

Unfortunately, this operation is not defined on all of S u b ( B ) , be- cause it does not deal with the case where Al n A2 is the empty set. But this problem can be solved, either by allowing the empty set to be considered as an algebra or by adjoining some new element to the set S u b ( B ) and defining Al A A2 to be this new element, whenever Al n A2 = 0.

The union Al U A2 of the universes of two subalgebras of B is in general not a subalgebra of B . But we can map the pair (Al, A2) to the subalgebra of B which is generated by the union Al U A2:

Altogether we obtain:

Theorem 1.1.7 For every algebra B , the algebra ( S u b ( B ) ; A , V ) is a lattice, called the subalgebra lattice of B .

Our second important algebraic construction is the formation of homomorphic images.


Definition 1.1.8 Let A = (A; (f,")iEI) and B = (B; (f?)iEI) be algebras of the same type r. Then a function h : A + B is called a homomorphism h : A + B of A into B if for all i E I we have

for all a l , . . . , ant E A. In the special case that ni = 0, this equation means that h(fk(0)) = fiB(0). Therefore the element designated by the nullary operation fk in A must be mapped to the corresponding element f: in B.

If h is surjective (onto), then B is called homomorphic image of A. If the function h is bijective, that is both one-to-one (injective) and "onto" (surjective), then the homomorphism h : A + B is called an isomorphism from A onto B. An injective homomorphism from A into B is also called an embedding of A into B.

A homomorphism h : A + A of an algebra A into itself is called an endomorphism of A, and an isomorphism h : A + A from A onto A is called an automorphism of A.

It is easy to see that the image B1 = h(A1) of a subalgebra Al of A under the homomorphism h is a subalgebra of B and that the preimage hpl(B') = A' of a subalgebra B' of h(A) C B is a subalgebra of A.

If X c A is a subset of the universe of an algebra A of type 7 , if (X)A is the subalgebra of A generated by X and if h : A + B is a homomorphism, then

If especially X is a generating system of A and if h : A + B is surjective, then h (X) generates B since

Let A, B and C be algebras of the same type, and let hl : A + B and h2 : B + C be homomorphisms, then h2 o hl : A + C is also a homomorphism, and when both hl and h2 are surjective, injective or bijective, then the composition has the same property.

8 1 Basic Concepts

Clearly, the identity mapping idA on the set A is always an automorphism of the algebra A. Since the composition operation o is associative, the set End(A) of all endomorphisms of an algebra A forms a monoid (End(A); 0, idA), the endomorphism monoid of A. Since the inverse mapping of an automorphism cp : A + A defines again an automorphism, the set Aut(A) of all automorphisms of A forms a group (Aut(A); o, idA), which we call the automorphism group of A.

Any homomorphism h : A + B goes out from the algebra A into the algebra B. We ask for an "internal" description of the homomorphic image inside B. Every function h : A + B from a set A onto a set B defines a partition of A into classes of elements having the same image. Partitions of a set define equivalence relations on that set where two elements are related to each other if and only if they belong to the same block of the partition. If A and B are the universes of two algebras A and B E Alg(r) and if h : A + B is a surjective homomorphism, then h can be compatible with an equivalence relation Q on A in the following sense:

Definition 1.1.9 Let A be a set, let 8 c A x A be an equivalence relation on A, and let f be an n-ary operation from On(A). Then f is said to be compatible with 8, or to preserve 8, if for all a1 , . . . , an,bl , . . . , b, E A ,

Definition 1.1.10 Let A = (A; ( f t ) i , I ) be an algebra of type 7 . An equivalence relation 8 on A is called a congruence relation on A if all the fundamental operations f: are compatible with 8. We denote by Con(A) the set of all congruence relations of the algebra A. For every algebra A = (A; (f:)i,I) the trivial equivalence relations

AA := { ( a , a ) I a E A} and VA = A x A

are congruence relations. An algebra which has no congruence relations except AA and VA is called simple.


For two congruence relations 81, Q2 E Con(A) the intersection Q1 nQ2 is again a congruence relation on A. This defines a binary operation

Simple examples show that the union Q1 U Q2 of two equivalence relations Q1, Q2 on the set A needs not to be an equivalence relation on A. To have a second binary operation on Con(A) we define at first the congruence relation defined by a binary relation on A.

Definition 1.1.11 Let A be an algebra, and let 8 be a binary relation on A. We define the congruence relation (Q)con(A) on A generated by Q to be the intersection of all congruence relations 8' on A which contain 8:

(Q)con(Aj : = n{Q' I 8' t Con(A) and Q C Q').

Again it can be seen that (Q)Con(A) has the three important properties of a closure operator:

Now we define the second binary operation on Con(A) by

v : Con(A) x Con(A) + Con(A) with (81~82) H (81 U ~ ~ ) C O ~ ( A )

and obtain:

Theo rem 1.1.12 For every algebra A the algebra (Con(A); A, V) is a lattice which i s called the congruence lattice of A.

If Q is a congruence relation on A, then we can partition the set A into blocks with respect to Q and obtain the quotient set A/Q. In a natural way, for each i t I, we define an ni-ary operation ftl* on the quotient set by

"I* : (A/Q)ni + A/Q f i

1 Basic Concepts

with

Of course, we have to verify that our operations are well-defined, that is, that they are independent on the representatives chosen. But this is exactly what the compatibility property of a congruence relation means and we obtain a new algebra A/Q :=

(AIQ; ( f t /* ) i , I ) , which is called the quotient algebra (or factor algebra) of A by 8. Actually, for every congruence relation Q the algebra A / 8 is a homomorphic image of A under the natural homomorphism defined by

n a t ( Q ) : A + A/Q w i t h a H [a]* for every a E A .

It is easy to check that n a t ( 8 ) is really a surjective homomorphism. So, for any congruence relation 8 E C o n ( A ) we obtain a homomorphism and it arises the question whether homomorphisms define congruence relations on A . This is also the case since we have:

Lemma 1.1.13 The kernel

ker h := { ( a , b) E A2 I h ( a ) = h ( b ) }

of any homomorphism h : A + B is a congruence relation o n A .

Suppose we have a homomorphism h : A + B. We have seen that ker h is a congruence on A, so we can form the quotient algebra A l k e r h, along with the natural homomorphism n a t ( k e r h) : A +

A l k e r h which maps the algebra A onto this quotient algebra. Now we have two homomorphic images of A: the original h(A) and the new quotient A l k e r h . What connection is there between these two homomorphic images? The answer to this question is given by the well-known Homomorphic Image Theorem

Theorem 1.1.14 (Homomorphic Image Theorem) Let h : A + B be a surjective homomorphism. T h e n there exists a unique isomorphism f from A l k e r h onto B with f o n a t ( k e r h) = h, that is, the following diagram commutes.

1.2 Direct and Subdirect Products

n n t (ker. h)\ f

Later on we need the sublattice Coni,,(A) of all fully invariant congruence relations on A .

Definition 1.1.15 Let A be an algebra of type 7. A congruence relation 8 E C o n ( A ) is called fully invariant if for all endomorphisms cp : A + A we have

(a, b) E 8 + (cp(a), cp(b)) E 8 for all a , b E A.

The congruence relations AA and VA are always fully invariant.

Proposition 1.1.16 The set Coni,,(A) of all fully invariant congruence relations of A forms a sublattice of C o n ( A ) .


In this section we shall examine another important construction, the formation of product algebras. Subalgebras or homomorphic images of a given algebra have cardinalities no larger than the cardinality of the given algebra. The formation of products, however, can lead to algebras with bigger cardinalities than those we started with. At first we consider the direct product of a family of algebras.

Definition 1.2.1 Let be a family of algebras of type r. The direct product n Slj of the Slj is defined as an algebra with

I € J

the carrier set

12 1 Basic Concepts

and the operations

for a,, . . ., ant in P; that is,

If for all j E J, Aj = A, then we usually write AJ instead of n 4. If J = 0, then A0 is defined to be the one-element (trivial) jtJ

algebra of type r. If J = {I, . . . , n), then the direct product can be written as A1 x . . x A,.

The projections of the direct product n Aj are the mappings jt J

pk : n Aj + Ale defined by (aj) jtJ ah-. jtJ

It is easy to check that the projections of the direct product are in fact surjective homomorphisms.

We recall the definition of the product (composition) Q1 o Q2 of two binary relations Q1, 82 on any set A:

Q1 0 Q2 := {(a, b) I 3c E A ((a, c) E Q2 A (c, b) E Q1)).

Two binary relations Q1,Q2 on A are called permutable, if Q1 o Q2 = Q2 o Q1.

We now consider a direct product of two factors. In this case we have two projection mappings, pl and p2, each of which has a kernel which is a congruence relation on the product, since pl , p2 are homomorphisms. These two kernels have special properties.

Lemma 1.2.2 Let A1, A2 be two algebras of type T and let Al x A2 be their direct product. Then:

(ii) ker pl o ker p2 = ker p2 o ker pl;


This lemma motivates the following definition:

Definition 1.2.3 A congruence 8 on A is called a factor congruence if there is a congruence 8' on A such that 8A8' = aA 8V8' = vA.

The pair (8,8') is called a pair of factor congruences on A.

Theorem 1.2.4 If (8,Q') i s a pair of factor congruences, t h e n A i s isomorphic wi th the direct product A/8 x A/8' (A " A/8 x A/V) under the i somorphism given by a H ([ale, [ale!).

Definition 1.2.5 An algebra A is called directly indecomposable, if whenever A " B1 x B2, either lBll = 1 or IB21 = 1.

It is well-known (see e.g. [37]) that there is up to isomorphism only one non-trivial directly indecomposable Boolean algebra, namely the two-element Boolean algebra ((0, 1); A, V , 1 , 0 , 1 ) . Cardinality considerations show that a countably infinite Boolean algebra cannot be isomorphic to a direct product of directly indecomposable algebras. But for finite algebras we have:

Theorem 1.2.6 Every finite algebra A i s isomorphic t o a direct product of directly indecomposable algebras.

This can be proved by induction on the cardinality of A. Our con- sideration shows that directly indecomposable algebras are not the "general building blocks" in the study of universal algebra. There- fore, we define another kind of products.

Definition 1.2.7 Let (Aj)j,J be a family of algebras of type T. A subalgebra B c n Aj of the direct product of the algebras Aj is

. I€ J

called a subdirect product of the algebras Aj, if for every projection mapping pk : n Slj + Ak we have

jeJ

pk(B) = Ak.

Examples for subdirect products are the diagonal AA = {(a, a ) I a E A) as well as any direct product. The lattice given by the Hasse diagram below

is the direct product of the two-element lattice C2 e b

and the three-element lattice C3 l 3

The sublattice C C C2 x C3 which is described by the diagram

l (4 3 )

is obviously a subdirect product of C2 and Cj.

It can easily be checked that for the subdirect product B = Aj j E J

of the family (Aj)jEJ the projection mappings p k : n Aj + Ak j E J

1.2 Direct and Subdirect Products 15

satisfy the equation n ker (p j I B) = AB. It turns out that this j t J

property of the kernels of the projection mappings can be used to characterize subdirect products, in the sense that any set of congruences on an algebra with these properties can be used to express the algebra as a subdirect product.

Theorem 1.2.8 Let A be a n algebra. Let { Q j I j E J } be a family of congruence relations o n A, which satisfy the equation n Q j =

.I€ J

AA. T h e n A is isomorphic to a subdirect product of the -algebras A/Bj, for j t J . I n particular, the mapping ~ ( a ) := ( [a ]Qi ) j t J defines a n embedding cp : A + n (A/Qj) , whose image p(A) is a

j € J

subdirect product of the algebras A/Qj.

We remark that the converse of this theorem is also true. If A is isomorphic to a subdirect product of a family (Aj)j, of algebras, then there exists a family of congruence relations on A whose intersection is the relation AA.

Algebras which cannot be expressed as a subdirect product of other smaller algebras, except in trivial ways, are called subdirectly irreducible.

Definition 1.2.9 An algebra A of type r is called subdirectly irreducible, if every family { Q j I j E J) of congruences on A, none of which is equal to AA, has an intersection which is different from AA. In this case, the conditions of Theorem 1.2.8 are not satisfied, and no representation of A as a subdirect product is possible.

It is easy to see that an algebra A is subdirectly irreducible if and only if AA has exactly one upper neighbour or cover in the lattice Con(A) of all congruence relations on A. Then the congruence lattice has the form shown in the diagram below.

16 1 Basic Concepts

Subdirect products do have the right property to act as the "general building blocks" in the study of universal algebra, as the following result of G. Birkhoff ( [ 8 ] ) shows.

Theorem 1.2.10 Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.

1.3 Term Algebras, Identities, Free Al- gebras

We need an appropriate language if we want to describe classes of algebras of the same type by logical expressions. This formal language is built up by variables from an n-element set X, =

{xl , . . . , x,), n > 1. The set X, is called an alphabet. We also need a set { fili E I ) of operation symbols, indexed by the set I. The sets X, and {fili E I ) have to be disjoint. To every operation symbol fi we assign a natural number ni > I , called the arity of fi. The sequence r = (ni)iE1 is called the type of the language. Now we define the terms of our type r, the "words" of our language.

Definition 1.3.1 Let n > I . The n-ary terms of type r are defined in the following inductive way:

(i) Every variable xi E X, is an n-ary term.

(ii) If t l , . . . , tni are n-ary terms and fi is an ni-ary operation symbol, then fi (tl , . . . , tni) is an n-ary term.

(iii) The set W,(X,) = W,(xl,. . . , x,) of all n-ary terms is the smallest set which contains xl , . . . , x, and is closed under finite application of (ii) .

It follows immediately from the definition that every n-ary term is also k-ary, for k > n.

Our definition does not allow nullary terms. This could be changed by adding a fourth condition to the inductive definition, stipulating that every nullary operation symbol of our type is an n-ary term. We could also extend our language to include a third set of symbols, to be used as constants or nullary terms.

1.3 Term Algebras, Identities, Free Algebras 17

Our definition of terms is inductive, based on the number op(t) of occurrences of operation symbols in a term. The operation symbol count op(t) is also inductively defined by

(i) op(xi) := 0, if xi EX,,

(ii) op(fi(t1,. . . ,t,%)) := C op(tj) + 1. j=1

To determine op(t) is one of the methods to measure the complexity of a term. Another common complexity measure is the depth of a term defined by the following steps:

(i) depth(xi) := 0, if xi E X,,

(ii) depth(fi(t1,. . . , t,,)) := max{depth(tl), . . . , depth(t,,)} + 1.

In a similar way the mindepth of a term can be defined if we replace the second step of the definition of depth by

mindepth(fi(tl,. . . , t,,)) := min{depth(tl), . . . , depth(t,%)} + 1.

Terms can be illustrated by tree diagrams, also called semantic trees. The semantic tree of the term t is defined as follows:

(i) If t = xi, then the semantic tree of t consists only of one vertex which is labelled with xi, and this vertex is called the root of the tree.

(ii) If t = f i( t l , . . . , t,%) then the semantic tree of t has as its root a vertex labelled with fi, and has ni edges which are incident with the vertex fi; each of these edges is incident with the root of the tree corresponding to one of the terms t l , . . . , t,% (ordered by 1 < 2 < 0 . < ni, starting from the left).

As an example we show the diagram of the term

m-solid varieties of algebras

Documents