roughness in mv-algebras

Information Sciences 180 (2010) 737–747

Contents lists available at ScienceDirect

Information Sciences

journal homepage: www.elsevier .com/locate / ins

Roughness in MV-algebras

S. Rasouli, B. Davvaz *

Department of Mathematics, Yazd University, Yazd, Iran

a r t i c l e i n f o

Article history:Received 20 February 2009Received in revised form 8 October 2009Accepted 2 November 2009

Keywords:Rough setLower approximationUpper approximationMV-algebraIdealSubalgebraRough subalgebra

0020-0255/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.ins.2009.11.008

* Corresponding author.E-mail address: [email protected] (B. Davvaz

a b s t r a c t

In this paper, by considering the notion of an MV-algebra, we consider a relationshipbetween rough sets and MV-algebra theory. We introduce the notion of rough ideal withrespect to an ideal of an MV-algebra, which is an extended notion of ideal in an MV-alge-bra, and we give some properties of the lower and the upper approximations in an MV-algebra.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

The theory of rough sets was introduced by Pawlak [29] in 1982, also see [26–28]. The theory of rough sets is an extensionof set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approxima-tions. A key concept in Pawlak rough set model is an equivalence relation. The equivalence classes are the building blocks forthe construction of the lower and upper approximations. The lower approximation of a given set is the union of all the equiv-alence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes whichhave a non-empty intersection with the set. The objects of the given universe U can be divided into three classes with respectto any subset A # U:

(1) the objects, which are definitely in A;(2) the objects, which are definitely not in A;(3) the objects, which are possibly in A.

The objects in class (1) form the lower approximation of A, and the objects in types (1) and (3) together form its upperapproximation. Some authors, for example, Iwinski [17], and Pomykala and Pomykala [31] have studied algebraic propertiesof rough sets. The lattice theoretical approach has been suggested by Iwinski [17]. Pomykala and Pomykala [31] showed thatthe set of rough sets forms a Stone algebra. Comer [7] presented an interesting discussion of rough sets and various algebrasrelated to the study of algebraic logic, such as Stone algebras and relation algebras. A natural question is what will happen ifwe substitute an algebraic system instead of the universe set. Biswas and Nanda [1] introduced the notion of rough sub-groups. Kuroki in [21], introduced the notion of a rough ideal in a semigroup, also see [34]. Davvaz in [8] introduced the

. All rights reserved.

).

http://dx.doi.org/10.1016/j.ins.2009.11.008

mailto:[email protected]

http://www.sciencedirect.com/science/journal/00200255

http://www.elsevier.com/locate/ins

738 S. Rasouli, B. Davvaz / Information Sciences 180 (2010) 737–747

notion of rough subrings (respectively ideal) with respect to an ideal of a ring, also see [11]. In [18], Kazanci and Davvazintroduced the notions of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in a ring and gave some prop-erties of such ideals. Rough modules have been investigated by Davvaz and Mahdavipour [14]. In [22], Leoreanu-Fotea andDavvaz introduced the concept of roughness in n-ary hypergroups, also see [19]. For more information about rough sets andalgebraic properties of rough sets, we refer to [9,10,12,13,15,16,25,30,32,39].

Chang in [4] introduced the notion of an MV-algebra to provide an algebraic proof of the Completeness theorem of infinitevalued Łukasiewicz propositional calculus. An MV-algebra A is an abelian monoid hA;0;�i equipped with an operation � suchthat ðx�Þ� ¼ x; x� 0� ¼ 0� and finally, ðx� � yÞ� � y ¼ ðy� � xÞ� � x. An example of an MV-algebra is given by the real unit inter-val [0,1] equipped with the operations x� ¼ 1� x and x� y ¼minð1; xþ yÞ. Valid equations yield new valid equations bysubstituting equals for equals. Chang’s Completeness theorem states that in this way one obtains from the above equationsevery valid equation in the MV-algebra [0,1]. Boolean algebra stand to Boolean logic as MV-algebra stand to Łukasiewiczinfinite valued logic. In order to see the approximations on MV-related structures, one can see [2,3].

In this paper, we consider an MV-algebra as a universal set and we shall introduce the notion of rough ideal with respectto an ideal of an MV-algebra, which is an extended notion of an ideal in an MV-algebra. We give some properties of the lowerand the upper approximations in an MV-algebra.

2. A brief excursion into MV-algebras and approximation spaces

2.1. MV-algebras

In this section first we recall the notion of an MV-algebra and then we review some definitions and properties which wewill need in the next section. Also, we will prove some properties in respect to convex subsets of a linearly ordered MV-alge-bra. Our main references in studying of MV-algebras are [4,6].

Definition 2.1.1. An MV-algebra is a structure ðM;�; �;0Þ, where � is a binary operation, � is a unary operation, and 0 is aconstant such that the following axioms are satisfied for any a; b 2 M:

(MV1) ðM;�; 0Þ is a commutative monoid,(MV2) ða�Þ� ¼ a,(MV3) 0� � a ¼ 0�,(MV4) ða� � bÞ� � b ¼ ðb� � aÞ� � a.

If we define the constant 1 :¼ 0� and the auxiliary operations �;_, and ^ by

a� b :¼ ða� � b�Þ�; a _ b :¼ a� ðb� a�Þ; a ^ b :¼ a� ðb� a�Þ;

then ðM;�;1Þ is a commutative monoid and the structure ðM;_;^;0;1Þ is a bounded distributive lattice. Also, we define thebinary operation � by x� y :¼ x� y�. Now, if we define x 6 y if and only if x ^ y ¼ x for each x; y 2 M, then according to [4],6 is an order relation over M. If the order relation 6, defined over M, is total, then we say that M is linearly ordered. We writenx instead of x� � � � � xðn-timesÞ. Also, we define the order of an element x, in symbols ordðxÞ, is the least integer m such thatmx ¼ 1. If no such integer m exists then ordðxÞ ¼ 1. We say MV-algebra M is locally finite if and only if, every element of Mdifferent from 0 has a finite order. Let X be a subset of an MV-algebra M. Chang in [4] has shown that every locally finite MV-algebra is linearly ordered. As usual, we say that X is an MV-subalgebra (for short, subalgebra) of M if and only if X is closedunder the MV-operations defined in M.

In an MV-algebra M, the distance function is

d : M �M ! M; dða; bÞ :¼ ða� � bÞ � ðb� � aÞ:

Proposition 2.1.2 [4]. Let M be an MV-algebra. Then the following properties hold for every x; y; z 2 M:

(i) dðx; yÞ ¼ dðy; xÞ,(ii) dðx; yÞ ¼ 0 if and only if x ¼ y,

(iii) dðx;0Þ ¼ x; dðx;1Þ ¼ x�,(iv) dðx�; y�Þ ¼ dðx; yÞ,(v) dðx; yÞ 6 ðdðx; zÞ � dðz; yÞÞ,

(vi) dðx� u; y� vÞ 6 ðdðx; yÞ � dðu;vÞÞ,(vii) x� x ¼ 0,

(viii) ðx� yÞ � y ¼ ðy� xÞ � x.

Let M be an MV-algebra and BðMÞ ¼ fx 2 M : x� x ¼ xg ¼ fx 2 M : x� x ¼ xg. Then according to [4], the systemðBðMÞ;�; �;0Þ is not only a subalgebra of M but is also the largest subalgebra of M which is at the same time a Booleanalgebra.

S. Rasouli, B. Davvaz / Information Sciences 180 (2010) 737–747 739

Definition 2.1.3. Let M be an MV-algebra and I be a non-empty subset of M. Then we say that I is an ideal if the followingconditions are satisfied:

(i) 0 2 I,(ii) x; y 2 I imply x� y 2 I,

(iii) x 2 I and y 6 x imply y 2 I.

Let IðMÞ denote the set of all ideals of M. Now let H be a non-empty subset of M. Then we set

hHi ¼ fx 2 M : x 6 h1 � � � � � hn;h1; . . . ;hn 2 H; n 2 Ng:

With the above notations, it is easy to check that hHi is the smallest ideal of M that contains H. As usual, a proper idealN 2 IðMÞ is called maximal if for every I 2 IðMÞ such that N # I, either N ¼ I, or I ¼ M. A proper ideal P 2 IðMÞ is called primewhenever x ^ y 2 P then either x 2 P, or y 2 P. The set of all prime ideals of an MV-algebra M is denoted by specðMÞ and theset of maximal ideals of M is denoted by MaxðMÞ. We can show each maximal ideal is a prime ideal so MaxðMÞ# specðMÞ. Theintersection of all maximal ideals, the radical M, is denoted by RadðMÞ. We say an MV-algebra M is simple if M has no anyideal expect 0 ¼ f0g and M. As a consequence we can show that an MV-algebra M is locally finite if and only if M is simple.As usual, every ideal I induces the congruence relation I .

Let M be an MV-algebra and I be an ideal of M. Then the following relation:

xIy dðx; yÞ 2 I:

is a congruence relation. It means that xIy implies x� zIy� z and x�Iy�. The equivalence class of x 2 M is denoted by ½x�I .A map f from an MV-algebra M to another MV-algebra M0 is called homomorphism if and only if the following conditions

are satisfied for every x; y 2 M, f ð0Þ ¼ 0, f ðx� yÞ ¼ f ðxÞ � f ðyÞ, f ðx�Þ ¼ f ðxÞ�.Now, suppose that M and M0 are two MV-algebras, f is a homomorphism from M to M0 and kerðf Þ ¼ f�1ð0Þ. Then f ðMÞ is a

subalgebra of M0 and kerðf Þ 2 IðMÞ.

Proposition 2.1.4 [6]. Let M be an MV-algebra, I an ideal of M and x; y 2 M. Then the following conditions are equivalent:

(i) ½x�I ¼ ½y�I ,(ii) x ¼ ðy� hÞ � k� for some h; k 2 I.

Definition 2.1.5. Let M be a linearly ordered MV-algebra and X be a subset of M. Then X is called convex if for every x; y 2 X
and z 2 M; x 6 z 6 y implies that z 2 X.
Proposition 2.1.6. Let I be an ideal of a linearly ordered MV-algebra M. Then ½x�I is convex for each x 2 M.

Proof. Let t; s 2 ½x�I and t 6 z 6 s. It is easy to check that if x 6 y then y� 6 x� and dðx; yÞ ¼ x� � y. Thus we have

dðt; zÞ � dðz; sÞ ¼ ðt� � zÞ � ðz� � sÞ 6 dðt; sÞ � dðt; sÞ

Since dðt; sÞ 2 I and I is an ideal, we conclude that dðt; zÞ � dðz; sÞ 2 I. On the other hand

dðz; xÞ 6 dðz; tÞ � dðt; xÞ � dðz; sÞ � dðs; xÞ 2 I:

So dðz; xÞ 2 I and it implies that z 2 ½x�I . Hence ½x�I is convex. h

Lemma 2.1.7. Let M be a linearly ordered MV-algebra and I be an ideal of M. If x 6 y and ½x�I – ½y�I , then for each t 2 ½x�I ands 2 ½y�I; t 6 s.

Proof. Suppose that there exist t 2 ½x�I and s 2 ½y�I such that s 6 t. We show that it is a contradiction. First, let t 6 y so weobtain s 6 t 6 y and by Proposition 2.1.6, t 2 ½y�I . It is a contradiction. Now, let y 6 t, thus x 6 y 6 t and by Proposition2.1.6, y 2 ½x�I . It is a contradiction again. Therefore for each t 2 ½x�I and s 2 ½y�I; t 6 s. h

Proposition 2.1.8 [4]. If M is a linearly ordered MV-algebra, then x� y ¼ x� z and x� z – 1 implies that y ¼ z.

2.2. Pawlak approximation spaces

Let U denote a finite and non-empty set called the universe. Let h # U � U be an equivalence relation on U. The pairApr ¼ ðU; hÞ is called a Pawlak approximation space. The equivalence relation h partitions the set U into disjoint subsets.Let U=h denote the quotient set consisting of all the equivalence classes of h. The empty set ; and the elements of U=h arecalled elementary sets. A finite union of elementary sets, i.e., the union of one or more elementary sets, is called a composedset [24]. The family of all composed sets is denoted by ComðAprÞ. It is a subalgebra of the Boolean algebra 2U formed by the


power set of U. A set which is a union of elementary sets is called a definable set [24]. The family of all definable sets is de-noted by Def ðAprÞ. For a finite universe, the family of definable sets is the same as the family of composed sets. A Pawlakapproximation space defines uniquely a topological space ðU;Def ðAprÞÞ, in which Def ðAprÞ is the family of all open and closedsets [29]. In connection to rough set theory there exist two views. The operator-oriented view interprets rough set theory asan extension of set theory with two additional unary operators. Under such a view, lower and upper approximations are re-lated to the interior and closure operators in topological spaces, the necessity and possibility operators in modal logic, andlower and upper approximations in interval structures. The set-oriented view focuses on the interpretation and character-ization of members of rough sets. Both operator-oriented and set-oriented views are useful in the understanding and appli-cation of the theory of rough sets. Yao in [37] studied both operator-oriented and set-oriented views.

Definition 2.2.1. For an approximation space ðU; hÞ, by a rough approximation in ðU; hÞ we mean a mappingApr : PðUÞ ! PðUÞ � PðUÞ defined for every X 2 PðUÞ by

AprðXÞ ¼ ðAprðXÞ;AprðXÞÞ

where AprðXÞ ¼ fx 2 U : ½x�h # Xg; AprðXÞ ¼ fx 2 U : ½x�h \ X – ;g. AprðXÞ is called a lower rough approximation of X in ðU; hÞ,where as AprðXÞ is called an upper rough approximation of X in ðU; hÞ, see [20,23,33,35,36,38].

Now, we give some examples of the lower and upper approximations theory applied to the MV-algebra theory.

Example 2.2.2. Let S7 ¼ f0;1=7;2=7; . . . ;6=7;1g. We define p=7þ q=7 :¼ minfðpþ qÞ=7;1g and ðp=7Þ� :¼ ð7� pÞ=7, thenðS7;þ; �;0Þ is an MV-algebra. Now, let h be an equivalence relation with the following equivalence classes:

E1 ¼ f0;3=7;4=7g;E2 ¼ f1=7;6=7g;E3 ¼ f2=7g;E4 ¼ f5=7g:

Let X :¼ f2=7;4=7g. Then AprðXÞ ¼ f2=7g and AprðXÞ :¼ f0;2=7;3=7;4=7g.

Definition 2.2.3. Let AprðXÞ ¼ ðAprðXÞ;AprðXÞÞ and AprðYÞ ¼ ðAprðYÞ;AprðYÞÞ be any two rough sets in the approximationspace ðU; hÞ. Then

(i) AprðXÞ t AprðYÞ ¼ ðAprðXÞ [ AprðYÞ;AprðXÞ [ AprðYÞÞ,(ii) AprðXÞ u AprðYÞ ¼ ðAprðXÞ \ AprðYÞ;AprðXÞ \ AprðYÞÞ,

(iii) AprðXÞ v AprðYÞ () AprðXÞ u AprðYÞ ¼ AprðXÞ.

When AprðXÞ v AprðYÞ, we say that AprðXÞ is a rough subset of AprðYÞ. Thus in the case of rough sets AprðXÞ andAprðYÞ; AprðXÞ v AprðYÞ if and only if AprðXÞ# AprðYÞ and AprðXÞ# AprðYÞ. This property of rough inclusion has all the prop-erties of set inclusion.

Example 2.2.4. Consider the real unit interval ½0;1� equipped with the operations x� ¼ x� 1 and x� y ¼ minð1; xþ yÞ. Thenð½0;1�;�; �;0Þ is an MV-algebra which is called the standard MV-algebra. Now, let h be an equivalence relation with thefollowing definition:

xhy if and only if x� y 2 Q:

Let X ¼ f1=n : n 2 Ng and Y ¼ f1=r : r 2 Rg. So we have AprðXÞ ¼ ;; AprðYÞ ¼ ;; AprðXÞ ¼ Q \ ½0;1� and AprðYÞ ¼ ½0;1�.Therefore, we have

AprðXÞ t AprðYÞ ¼ ð;; ½0;1�Þ;AprðXÞ u AprðYÞ ¼ ð;;Q \ ½0;1�Þ:

Also, we obtain AprðXÞ v AprðYÞ.The rough complement of AprðXÞ denoted by AprcðXÞ is defined by

AprcðXÞ ¼ ðU n AprðXÞ;U n AprðXÞÞ:

Example 2.2.5. Let M ¼ f0; x1; x2;1g. Consider the following tables:

Then ðM;�; �;0Þ is an MV-algebra.


Let h be an equivalence relation with the following equivalence classes:

E1 ¼ f0; x1g;E2 ¼ fx2;1g;

Let X :¼ f0; x1; x2g. Then AprcðXÞ ¼ ð;; E2Þ.Let ðU; hÞ be an approximation space and X be a non-empty subset of U.

(i) If AprðXÞ ¼ AprðXÞ, then X is called definable with respect to h.(ii) If AprðXÞ ¼ ;, then X is called empty interior with respect to h.

(iii) If AprðXÞ ¼ U, then X is called empty exterior with respect to h.

Let ðU; hÞ be an approximation space. Clearly, ;, U and each element of Def ðAprÞ are definable with respect to h.

Example 2.2.6. In Example 2.2.2, X is neither definable nor empty interior nor empty exterior with respect to h. In Example2.2.4, X is neither definable nor empty exterior but it is empty interior.

3. Approximations in MV-algebras

In mathematics and computer science, many valued logics have many applications. Indeed, it is well known that Booleanalgebras have an important role in applied mathematics so their generalizations, i.e., MV-algebras.

3.1. Rough ideals

Throughout this paper M is an MV-algebra. Let I be an ideal of M and X be a non-empty subset of M. Then the sets

AprIðXÞ ¼ fx 2 Mk½x�I # Xg; AprIðXÞ ¼ fx 2 Mj½x�I \ X – ;g
are called, respectively, the lower and upper approximations of the set X with respect to the ideal I.
Let I be an ideal of M, for a; b 2 M we say a is congruent to b module of I, written as a bðmodIÞ if dða; bÞ 2 I. Therefore,when U ¼ M and h is the induced congruence relation by ideal I, then we use the pair ðM; IÞ instead of the approximationspace ðU; hÞ. Also, in this case we use the symbols AprIðXÞ and AprIðXÞ instead of AprðXÞ and AprðXÞ. Furthermore, ifAprIðXÞ ¼ AprIðXÞ, we say X is definable respect to I.

Proposition 3.1.1. For every approximation space ðM; IÞ, every subsets X;Y � M and each x 2 X, we have:

(1) AprIðXÞ# X # AprIðXÞ,(2) M and ; are definable sets respect to any ideal,(3) AprIðXÞ; AprIðXÞ and ½x�I are definable sets respect to I,(4) If X # Y , then AprIðXÞ v AprIðYÞ,(5) AprIðXÞ ¼ ðAprIðXcÞÞc ,(6) AprIðXÞ ¼ ðAprIðXcÞÞc ,(7) AprIðX \ YÞ ¼ AprIðXÞ \ AprIðYÞ,(8) AprIðX \ YÞ# AprIðXÞ \ AprIðYÞ,(9) AprIðX [ YÞ AprIðXÞ [ AprIðYÞ,

(10) AprIðX [ YÞ ¼ AprIðXÞ [ AprIðYÞ,(11) AprMðXÞ ¼ ð;;MÞ if X – M,(12) Apr0ðXÞ ¼ ðX;MÞ.

Proof. The proof is similar to the proof of Theorem 2.1 of [21]. h

The following example shows that the converse of 8 and 9 in the Proposition 3.1.1 is not true.

Example 3.1.2. Let M ¼ f0; x1; x2; x3; x4;1g. Consider the following tables:


Then ðM;�; �;0Þ is an MV-algebra. Now, it is easy to check that I ¼ f0; x2g is an ideal. Let X ¼ f0; x2; x4;1g andY ¼ f0; x1; x3; x4g are subsets of M. Then the equivalence classes are ½0�I ¼ ½x2�I ¼ I; ½x1�I ¼ ½x4�I ¼ fx1; x4g and½1�I ¼ ½x3�I ¼ fx3;1g. Therefore, we have

AprIðXÞ ¼ f0; x2g;AprIðYÞ ¼ fx1; x4g;AprIðX [ YÞ ¼ M;

AprIðXÞ ¼ M;

AprIðYÞ ¼ M;

AprIðX \ YÞ ¼ f0; x1; x2; x3g;

and so

AprIðX [ YÞ� AprIðXÞ [ AprIðYÞ

and

AprIðXÞ \ AprIðYÞ� AprIðX \ YÞ:

If M is a simple MV-algebra, then we know that M has only two ideals 0 and M. Therefore, by Proposition 3.1.1, we can obtainevery information about the lower and upper approximations in M. Also, we know that M is locally finite if and only if M issimple. Thus, we let M is not locally finite.

If X is a non-empty subset of an MV-algebra M, we let X� :¼ fx� : x 2 Xg. It is easy to check that for every X;Y # M if wehave X # Y , then X�# Y�.

Theorem 3.1.3. Let I be an ideal of M and X be a non-empty subset of M. Then

(i) AprIðXÞ� ¼ AprIðX�Þ,(ii) AprIðXÞ� ¼ AprIðX�Þ.

Proof

(i) Let x 2 AprIðXÞ� so x� 2 AprIðXÞ. Therefore, ½x�I \ X – ; so there exists y 2 ½x�I \ X. We have dðz; x�Þ 2 I and z 2 X. ByProposition 2.1.2, dðz�; xÞ 2 I and z� 2 X�, hence z� 2 ½x��I \ X� and this implies that x 2 AprIðX�Þ. Conversely, letx 2 AprIðX�Þ so ½x�I \ X� – ;. Suppose that z 2 ½x�I \ X�. Then z� 2 ½x��I \ X. Similarly, x 2 AprIðXÞ� and this provesthe part (i).

(ii) The proof is similar to the part (i). h

Proposition 3.1.4. Let M be a linearly ordered MV-algebra, I be an ideal of M and X be a convex subset of M. Then AprIðXÞ andAprIðXÞ are convex.

Proof. We show that AprIðXÞ is convex. Let x; y 2 AprIðXÞ and x 6 z 6 y. We must show ½z�I # X, so let t 2 ½z�I . By Lemma2.1.7, for all v 2 ½x�I and w 2 ½y�I we have v 6 t 6 w and since ½x�I; ½y�I # X and X is convex. Hence the result holds.

Now, let x; y 2 AprIðXÞ and x 6 z 6 y. Therefore ½x�I \ X–; and ½y�I \ X – ;, so there exist v 2 ½x�I \ X and w 2 ½y�I \ X.Now, let t 2 ½z�I , by Lemma 2.1.7, we have v 6 t 6 w so t 2 X. This implies that AprIðXÞ is convex. h

Let X be a non-empty subset of an MV-algebra M, and X? be the annihilator of X in M defined byX? ¼ fa 2 M : a ^ x ¼ 0; 8x 2 Xg. Notice that X? is a proper ideal of M. If X ¼ fxg, then we write x? for X?.

Proposition 3.1.5. Let I be an ideal of M and X be a non-empty subset of M. Then

(1) AprIðX?Þ# AprIðXÞ?,(2) AprIðXÞ? # AprIðX?Þ,(3) AprIðXÞ? # AprIðXÞ?.

Proof. By Proposition 3.1.1, we have AprðXÞ# X. It is easy to see that if X # Y then Y?# X?, so by this we can takeX?# AprðXÞ? and again by Proposition 3.1.1, we obtain AprðX?Þ# AprðXÞ?. Similarly, we can prove (2) and (3). h

Notice that the inclusion symbols # in Proposition 3.1.5, may not be replaced by an equal sign, as the next example shows.

Example 3.1.6. Consider M ¼ f0; x1; x2; x3; x4;1g as the MV-algebra in Example 2.2.5. Let X ¼ f0; x2; x4;1g and Y ¼ f0; x1; x3gbe subsets of M and I ¼ f0; x2g the ideal of M. It is easy to check that X? ¼ f0g and Y? ¼ f0; x2g, so we haveAprIðX?Þ ¼ ;;AprIðXÞ? ¼ f0; x1g;AprIðXÞ? ¼ f0g;AprIðY?Þ ¼ f0; x2g, and AprIðYÞ? ¼ f0g, so

AprIðXÞ? � AprIðX?Þ;


AprIðY?Þ� AprIðYÞ?;AprIðXÞ? � AprIðXÞ?:

3.2. Some properties of approximations in MV-algebras

Let X and Y be non-empty subsets of M. Then we have

X þ Y ¼ fa 2 M : a 6 x� y; x 2 X; y 2 Yg:

If either X or Y are empty, then we define X þ Y ¼ ;. Clearly, X þ Y ¼ Y þ X for every X;Y # M. If I and J are two subalgebras orideals of an MV-algebra M, we can show that I þ J is the smallest ideal such that contains I and J. In fact I þ J is the ideal gen-erated by I [ J. Moreover, if I, J and K are three ideals of M such that I # K and J # K then we obtain I þ J # K .

Proposition 3.2.1. Let I be an ideal of an MV-algebra M and X; Y be non-empty subsets of M. ThenAprIðX þ YÞ# AprIðXÞ þ AprIðYÞ. In particularly, if M is linearly ordered, then

AprIðX þ YÞ ¼ AprIðXÞ þ AprIðYÞ:

Proof. Let a 2 AprIðX þ YÞ so ½a�I \ ðX þ YÞ–;. Thus there exists b 2 ½a�I \ ðX þ YÞ and consequently, there exist x 2 X andy 2 Y such that b 6 x� y. On the other hand, since ½a�I ¼ ½b�I , by Proposition 2.1.4, there are h; k 2 I such thata ¼ ðb� hÞ � k�. It can be concluded that

a ¼ ðb� hÞ � k� 6 b� h 6 ðx� hÞ � y:

Clearly, ½x� h�I ¼ ½x�I; x 2 AprIðXÞ and y 2 AprIðYÞ hence a 2 AprIðXÞ þ AprIðYÞ.Now, let M be linearly ordered and a 2 AprIðXÞ þ AprIðYÞ, so there exist x 2 AprIðXÞ and y 2 AprIðYÞ such that a 6 x� y.

Let t 2 ½x�I \ X; s 2 ½y�I \ Y and z 2 ½a�I . By Lemma 2.1.7, z 6 s� t and this implies that z 2 X þ Y . Therefore a 2 AprIðX þ YÞ,and the equality holds. h

Lemma 3.2.2. Let I be an ideal of an MV-algebra M and X be a non-empty subset of M. Then X is definable if and only ifAprIðXÞ ¼ X or AprIðXÞ ¼ X.

Proof. Obviously, if X is definable, we have AprIðXÞ ¼ X and AprIðXÞ ¼ X. Now, let AprIðXÞ ¼ X. Suppose that x 2 AprIðXÞ so½x�I \ X – ;. There exists y 2 ½x�I \ X, thus ½x�I ¼ ½y�I and y 2 X ¼ AprIðXÞ. This implies that ½x�I # X and so x 2 AprIðXÞ.

Now, let AprIðXÞ ¼ X. Suppose that x 2 AprIðXÞ and y 2 ½x�I . Thus, ½x�I \ X – ; and since ½x�I ¼ ½y�I , we have ½y�I \ X – ; and itimplies that y 2 AprIðXÞ ¼ X, so x 2 AprIðXÞ and X is definable. h

Properties 3.2.3. Let M be an MV-algebra, I be an ideal of M and X;Y be subsets of M such that X þ I ¼ X or Y þ I ¼ Y. Then X þ Yis a definable set respect to I. In particular, if X is an arbitrary subset of M, then X þ I is definable set with respect to I.

Proof. By Lemma 3.2.2, it is sufficient to prove that AprIðX þ YÞ ¼ X þ Y . Let X þ I ¼ X and a 2 AprIðX þ YÞ. Similar to Prop-osition 3.2.1, we can find some h 2 I such that a 6 ðx� hÞ � y. Clearly, x� h 2 X þ I ¼ X, so a 2 X þ Y . It means X þ Y is defin-able set respect to I. Furthermore, it is easy to see that I þ I ¼ I, so X þ I is definable set respect to I for each subset X of M. h

Proposition 3.2.4. Let I be an ideal of an MV-algebra of M and X;Y be non-empty subsets of M. ThenAprIðXÞ þ AprIðYÞ# AprIðX þ YÞ.

Proof. Suppose that a 2 AprIðXÞ þ AprIðYÞ so there exist x 2 AprIðXÞ and y 2 AprIðYÞ such that a 6 x� y. Now, let b 2 ½a�I ,thus by Proposition 2.1.4, there exist h; k 2 I such that b ¼ ða� hÞ � k�, so we have

b ¼ ða� hÞ � k� 6 ða� hÞ 6 ðx� hÞ � y:

Also, we have x� h 2 ½x� h�I ¼ ½x�I # X and y 2 Y . Hence b 2 X þ Y , this implies that a 2 AprIðX þ YÞ. h

Notice that the following examples show that we can not replace the inclusion symbol # by an equal sign in Proposition3.2.4.

Example 3.2.5

(i) Consider M ¼ f0; x1; x2; x3; x4;1g as the MV-algebra in Example 2.2.5. Let X ¼ f0g and Y ¼ fx4g be subsets of M andI ¼ f0; x2g be the ideal of M. We have X þ Y ¼ f0; x1; x2; x4g, AprIðXÞ ¼ f0g and AprIðYÞ ¼ ;, thereforeAprIðX þ YÞ ¼ f0; x1; x2; x4g is not a subset of AprIðXÞ þ AprIðYÞ ¼ ;.


(ii) Let M be a linearly ordered MV-algebra such that it is not locally finite and I – 0 be a proper ideal of M. Let X ¼ f0g andY ¼ f1g. Clearly, X þ Y ¼ M so AprIðX þ YÞ ¼ M, but one can see that AprIðXÞ þ AprIðYÞ ¼ ;.

Lemma 3.2.6. Let I; J be two ideals of M such that I � J and let X be a non-empty subset of M. Then

(i) AprJðXÞ# AprIðXÞ,(ii) AprIðXÞ# AprJðXÞ.

Proof. It is straightforward. h

Proposition 3.2.7. Let I; J be two ideals of an MV-algebra M and X be a non-empty subset of M. If X # BðMÞ or M is linearlyordered, then

AprIþJðXÞ# AprIðXÞ þ AprJðXÞ:

Proof. Assume that X is a subset of M such that X # BðMÞ. Let x 2 AprIþJðXÞ, so ½x�IþJ \ X–;. Thus there exists s 2 ½x�IþJ \ Xwhich implies that dðs; xÞ 2 I þ J and s 2 X, hence by Proposition 2.1.4, there exist h; k 2 I þ J such that x ¼ ðs� hÞ � k� andalso there exist i 2 I and j 2 J such that h 6 i� j. Since X # BðMÞ, we have

x 6 s� h 6 ðs� iÞ � ðs� jÞ:
On the other hand, ½s� i�I ¼ ½s�I and ½s� j�J ¼ ½s�J . Therefore x 2 AprIðXÞ þ AprJðXÞ.
Now, if M is linearly ordered, then either I # J or J # I. Suppose that I # J, so I þ J ¼ J and by Lemma 3.2.6, we haveAprJðXÞ# AprIðXÞ þ AprJðXÞ. h

The following example shows that in Proposition 3.2.7, the symbol inclusion can be proper.

Example 3.2.8

(i) Let M ¼ f0; x1; x2; x3; x4; x5; x6;1g. Consider the following tables:

Then ðM;�; �; 0Þ is an MV-algebra such that BðMÞ ¼ M. It is easy to check that I ¼ f0; x1g and J ¼ f0; x2g are two idealsof M and obviously I þ J ¼ f0; x1; x2; x3g that it is an ideal of M too. We have ½0�I ¼ I; ½x2�I ¼ fx2; x3g; ½x4�I ¼fx4; x5g; ½1�I ¼ I� and ½0�J ¼ J; ½x1�I ¼ fx1; x3g; ½x4�I ¼ fx4; x6g; ½1�J ¼ J�, also ½0�IþJ ¼ I þ J and ½1�IþJ ¼ ðI þ JÞ�. Suppose that

X ¼ fx4; x6g, we have AprIþJðXÞ ¼ I þ J; AprIðXÞ ¼ fx4; x5; x6;1g; AprJðXÞ ¼ fx4; x6g and AprIðXÞ þ AprJðXÞ ¼ M, therefore

AprIþJðXÞ$AprIðXÞ þ AprJðXÞ.
(ii) Let M be a linearly ordered MV-algebra, 0 ¼ f0g be the ideal of M and t – 0 be an element of M such that ordðtÞ– 2. By
Proposition 3.2.3, t þ 0 is a definable set with respect to ideal 0, so we have t þ 0 # t þ t þ 0. Now, we claim thatt � t R t þ 0. Assume that t � t 2 t þ 0 so there exist s 6 t that t � t 6 s. By Proposition 2.1.8, we obtain t ¼ 0 andit is a contradiction. Hence, it implies that AprIðXÞ þ AprJðXÞ is not a subset of AprIþJðXÞ.

Proposition 3.2.9. Let I; J be two ideals of an MV-algebra M and X be a non-empty subset of M. Then

AprIþJðXÞ# AprIðXÞ þ AprJðXÞ:

Furthermore, if a 2 AprIðXÞ þ AprJðXÞ, then we obtain ½a�IþJ # AprIðXÞ þ AprJðXÞ.

Proof. Let a 2 AprIþJðXÞ, so ½a�IþJ # X. We know that a 6 a� a and ½a�I; ½a�J # ½a�IþJ hence a 2 AprIðXÞ þ AprJðXÞ.Now, suppose that a 2 AprIðXÞ þ AprJðXÞ. There exist x1 2 AprIðXÞ and x2 2 AprJðXÞ such that a 6 x1 � x2. Suppose that

b 2 ½a�IþJ so there exist h; k 2 I þ J such that b ¼ ða� hÞ � k�. Also, there are i 2 I and j 2 J such that h 6 i� j. Then we have

b 6 a� h 6 ðx1 � iÞ � ðx2 � jÞ;

but ½x1 � i�I ¼ ½x1�I and ½x2 � j�J ¼ ½x2�J . This implies that ½a�IþJ # AprIðXÞ þ AprJðXÞ. h


Corollary 3.2.10. By notations of Proposition 3.2.9, if X is an ideal of M, then we obtain AprIþJðXÞ ¼ AprIðXÞ þ AprJðXÞ.

Proof. Clearly, if X is an ideal of M, we can show easily, AprIðXÞ þ AprJðXÞ# X. Now, let a 2 AprIðXÞ þ AprJðXÞ so by Proposition3.2.9, ½a�IþJ # AprIðXÞ þ AprJðXÞ# X. It proves the corollary. h

In the following example we show that AprIðXÞ þ AprJðXÞ can be not a subset of AprIþJðXÞ.

Example 3.2.11

(i) Consider M ¼ f0; x1; x2; x3; x4; x5; x6;1g as the MV-algebra in Example 3.2.8. Let I ¼ f0; x1g and J ¼ f0; x2g be two idealsof M and X ¼ f0; x1; x2; x3; x4; x5g be a subset of M. We have AprIþJðXÞ ¼ f0; x1; x2; x3g but AprIðXÞ ¼ f0; x1; x2; x3; x4; x5gand AprJðXÞ ¼ f0; x1; x2; x3g. Therefore, AprIðXÞ þ AprJðXÞ ¼ M.

(ii) Consider Example 3.2.8 part (ii). Since t þ 0 is a definable set respect to 0, exactly we can obtain AprIðXÞ þ AprJðXÞ is nota subset of AprIþJðXÞ.

3.3. Connections between special ideals, subalgebras and homomorphisms

Now, we want to see which is the intersection of all lower approximation of a set X with respect to all prime ideals.According to [5] the intersection of all prime ideals is {0}.

Theorem 3.3.1. Let X be a non-empty subset of M. Then
\
P2specðMÞAprPðXÞ ¼ 0:

Proof. Let fIaga2A be a family of ideals of M. Then by Proposition 3.2.7, we can conclude thatT

a2AAprIa ðXÞ# AprTa2A

IaðXÞ. So

we have

\

P2specðMÞAprPðXÞ# Apr T

P2specðMÞPðXÞ

and we know thatT

P2specðMÞP ¼ 0. So clearly the result holds. h

Clearly, if I; J are ideals of an MV-algebra M, then I \ J is an ideal too. So by Proposition 3.2.1, it is easy to see that

AprIðXÞ \ AprJðXÞ# AprI\JðXÞ and AprI\JðXÞ# AprIðXÞ \ AprIðXÞ:

Notice that under some conditions the equality holds:

Proposition 3.3.2. Let I; J be two ideals of an MV-algebra M and X be a non-empty subset of M. Then

(i) If X is an ideal of M and I; J # X, or M is linearly ordered, then AprIðXÞ \ AprJðXÞ ¼ AprI\JðXÞ;(ii) If X is definable respect to I or J, or M is linearly ordered then AprI\JðXÞ ¼ AprIðXÞ \ AprIðXÞ.

Proof

(i) Assume that x is an arbitrary element of AprI\JðXÞ. Then x 2 X. Now, let y 2 ½x�I so dðx; yÞ 2 X. Since X is an ideal, we havedðx; yÞ � x 2 X, on the other hand

y 6 2ðx� � yÞ � y ¼ dðx; yÞ � x

and this means that y 2 X, which implies that x 2 AprIðXÞ. Similarly, we obtain x 2 AprJðXÞ and so x 2 AprIðXÞ \ AprJðXÞ.Furthermore, if M is linearly ordered then I # J or J # I. By this the proof of the theorem is straightforward.

(ii) First, assume that X is definable respect to I, so AprIðXÞ ¼ X. Therefore, we have AprIðXÞ \ AprJðXÞ ¼X \ AprJðXÞ ¼ X # AprI\JðXÞ and it proves the theorem. Moreover, if M is linearly ordered, then obviously equalityholds. h

Let I be an ideal of M and AprIðAÞ ¼ ðAprIðAÞ;AprIðAÞÞ be a rough set in the approximation space ðM; IÞ. If AprIðAÞ and AprIðAÞare ideals (resp. subalgebras) of M, then we call AprIðAÞ a rough ideal (resp. subalgebra). Notice that a rough subalgebra iscalled a rough MV-algebra too.

Proposition 3.3.3. Let M be an MV-algebra and I be an ideal of M.

(i) If X is a subalgebra of M, then AprIðXÞ is a subalgebra too.(ii) In particular, if M is a linearly ordered MV-algebra and J is an ideal of M, then AprIðJÞ is an ideal of M.


Proof

(i) Let x; y 2 AprIðXÞ. Then ½x�I \ X – ; and ½y�I \ X – ;, so there exist x1 2 ½x�I \ X and y1 2 ½y�I \ X. Since X is a subalgebra ofM we have x1 � y1 2 X and x1 � y1 2 ½x�I � ½y�I ¼ ½x� y�I , therefore x� y 2 AprIðXÞ. Now, suppose that x 2 AprIðXÞ. Letx1 2 ½x�I \ X, so we have dðx1; xÞ 2 I and x1 2 X. By Proposition 2.1.2, dðx�1; x�Þ 2 I and since X is a subalgebra, x�1 2 X.Hence the result holds.

(ii) Clearly, 0 2 AprIðJÞ. Assume that x 2 AprIðJÞ and y 6 x. Therefore ½x�I \ J – ; so there exists x1 2 ½x�I \ J. By Lemma 2.1.7,for each y1 2 ½y�I we have y1 6 x1 and since J is an ideal, we obtain y1 2 J. Hence obviously, y 2 AprIðJÞ. Similar to part(i), we can show AprIðJÞ is closed under the binary operation �. It can be concluded that AprIðJÞ is an ideal. h

Proposition 3.3.4. Let I and J be two ideals of M. Then AprIðJÞ is an ideal when I # J and J is not empty interior. Furthermore, if M islinearly ordered then J is definable or AprIðJÞ ¼ ð0;0Þ.

Proof. Suppose that J is not empty interior and I # J. Let x 2 AprIðJÞ and y 6 x. We have ½x�I # J and since x 2 ½x�I and J is anideal, we obtain y 2 J. Now, we must show that ½y�I # J. Suppose that z 2 ½y�I . Then dðz; yÞ 2 I so dðz; yÞ 2 J, it implies thatz 2 ½y�J ¼ J, so y 2 AprIðJÞ. Now, let x; y 2 AprIðJÞ so ½x�I # J and ½y�I # J. Let z 2 ½x� y�I , hence dðz; x� yÞ 2 J since x� y 2 J weobtain z 2 ½x� y�J ¼ J, therefore x� y 2 AprIðJÞ. It proved that AprIðJÞ is an ideal.

Now, suppose that M is linearly ordered and I; J be two ideals of M. Clearly, I # J or J � I. First let I # J, in this case we proveJ is definable. According to Lemma 3.2.6, we show AprIðJÞ ¼ J. Let x 2 J and y 2 ½x�I . If y 6 x then, since J is an ideal, y 2 J.Suppose that x < y so dðx; yÞ ¼ y� x is an element of J, also x 2 J. Therefore by Proposition 2.1.2, we have

y 2 ðx� yÞ � y ¼ ðy� xÞ � x 2 J;

so y 2 J and it says x 2 AprIðJÞ. Hence AprIðJÞ ¼ J and it means that J is definable.Now, let J # I. We must show both AprIðJÞ ¼ 0 and AprIðJÞ ¼ 0. First, suppose that x 2 AprIðJÞ. Thus ½x�I # J so by hypothesis

½x�I ¼ ½0�I ¼ 0 hence x ¼ 0. Now, let x 2 AprIðJÞ thus ½x�I \ J – ;. There exists y 2 ½x�J \ J which implies that dðx; yÞ 2 I and y 2 J.Since J � I we have y 2 I and ½x�I ¼ ½y�I ¼ ½0�I , then AprIðJÞ ¼ 0. In brief AprIðJÞ ¼ ð0;0Þ. h

Similarly, if I is an ideal and J is a subalgebra of M such that I # J and I is not empty interior, then AprIðJÞ is a subalgebra of M.

Theorem 3.3.5. Let M and M0 be two MV-algebras and f : M ! M0 an MV-algebra homomorphism. Then

(1) If X is a non-empty subset of M, then

f ðAprker f ðXÞÞ ¼ f ðXÞ:

(2) If f is onto, then X is a non-empty subset of M and I is an ideal of M contained ker f , then

f ðAprIðXÞÞ ¼ Aprf ðIÞðf ðXÞÞ:

Proof

(1) Since X # Aprker f ðXÞ, it follows that f ðXÞ# f ðAprker f ðXÞÞ. Conversely, let y 2 f ðAprker f ðXÞÞ. Then there exists x 2 Aprker f ðXÞsuch that f ðxÞ ¼ y. So we have ½x�ker f \ X – ;. Let a 2 ½x�ker f \ X. We have dða; xÞ 2 ker f and a 2 X. Since f is an MV-alge-bra homomorphism we take dðf ðaÞ; f ðxÞÞ ¼ 0, so by Proposition 2.1.2, we have f ðaÞ ¼ y. It proved the theorem.

(2) Let y 2 f ðAprIðXÞÞ. There exists x 2 AprIðXÞ such that y ¼ f ðxÞ. Since ½x�I \ X – ;, suppose that z 2 ½x�I \ X so we havedðx; zÞ 2 I and z 2 X. Now, since f is a homomorphism we have dðf ðxÞ; f ðzÞÞ 2 f ðIÞ and f ðzÞ 2 f ðXÞ. It means that½f ðxÞ�f ðIÞ \ f ðXÞ – ; hence y 2 Aprf ðIÞðf ðXÞÞ.

Conversely, let y 2 Aprf ðIÞðf ðXÞÞ, so ½y�f ðIÞ \ f ðXÞ– ;. Suppose that z 2 ½y�f ðIÞ \ f ðXÞ. It implies that dðz; yÞ 2 f ðIÞ and z 2 f ðXÞ.Since f is onto, there exists x 2 M such that y ¼ f ðxÞ and also there exist t 2 X and s 2 I which z ¼ f ðtÞ and dðf ðtÞ; f ðxÞÞ ¼ f ðsÞ sof ðdðt; xÞÞ ¼ f ðsÞ and it implies that f ðdðt; xÞ � sÞ ¼ 0. Hence dðt; xÞ � s 2 I. Now, by Proposition 2.1.2, we have

dðt; xÞ 6 ðs� dðt; xÞÞ � dðt; xÞ ¼ ðdðt; xÞ � sÞ � s;

and we know that ððdðt; xÞ � sÞ; sÞ 2 I2 and I is an ideal so dðt; xÞ 2 I hence t 2 ½x�I . Now, we conclude that ½x�I \ X–; sox 2 AprIðXÞ. This proves the theorem. h

Example 3.3.6. Consider M ¼ f0; x1; x2; x3; x4; x5; x6;1g as the MV-algebra in Example 3.2.8. Also, let N ¼ f0; y1; y2;1g. Con-sider the following tables:


The map f : M ! N given by f ð0Þ ¼ 0; f ðaÞ ¼ y1; f ðx1Þ ¼ y1; f ðx2Þ ¼ y2; f ðx3Þ ¼ 1; f ðx4Þ ¼ 0; f ðx5Þ ¼ y1; f ðx6Þ ¼ y2 andf ð1Þ ¼ 1 defines a homomorphism. We obtain ker f ¼ f0; x4g. Suppose that X ¼ fx1; x5; x6g. Then f ðXÞ ¼ fy1; y2g; Aprker f ðXÞ¼ fy1g and f ðAprker f ðXÞÞ ¼ fy1g.

4. Conclusion

We introduced a new kind of approximations, that is the universe of objects is endowed with an MV-algebra structureand an equivalence relation is defined with respect to the notion of an ideal. Some properties of the approximations are thenderived. This approach is different from the usual one on abstract algebras (including MV) where objects of the algebra areapproximated through objects of the algebra, see [2,3].

Acknowledgement

The authors are highly grateful to referees for their valuable comments and suggestions for improving the paper.

References

[1] R. Biswas, S. Nanda, Rough groups and rough subgroups, Bulletin of the Polish Academy of Sciences – Mathematics 42 (1994) 251–254.[2] G. Cattaneo, R. Giuntini, R. Pilla, BZMVdM algebras and Stonian MV-algebras (applications to fuzzy sets and rough approximations), Fuzzy Sets and

Systems 108 (1999) 201–222.[3] G. Cattaneo, D. Ciucci, Algebraic structures for rough sets, LNCS – Transactions on Rough Sets II 3135 (2004) 218–264.[4] C.C. Chang, Algebraic analysis of many valued logics, Transactions of the American Mathematical Society 88 (1958) 467–490.[5] C.C. Chang, A new proof of the completeness of the Łukasiewicz axioms, Transactions of the American Mathematical Society 93 (1959) 74–80.[6] R. Cignoli, I. D’Ottaviano, D. Mundici, Algebraic Foundations of Many-Valued-Reasoning, Kluwer, Dorderecht, 2000.[7] S.D. Comer, On connections between information systems, rough sets and algebraic logic, Algebraic Methods in Logic and Computer Science 28 (1993)

117–124.[8] B. Davvaz, Roughness in rings, Information Sciences 164 (2004) 147–163.[9] B. Davvaz, A new view of approximations in Hv -groups, Soft Computing 10 (11) (2006) 1043–1046.

[10] B. Davvaz, Approximations in Hv -modules, Taiwanese Journal of Mathematics 6 (2002) 499–505.[11] B. Davvaz, Roughness based on fuzzy ideals, Information Sciences 176 (2006) 2417–2437.[12] B. Davvaz, Rough subpolygroups in a factor polygroup, Journal of Intelligent and Fuzzy Systems 17 (6) (2006) 613–621.[13] B. Davvaz, A short note on algebraic T-rough sets, Information Sciences 178 (2008) 3247–3252.[14] B. Davvaz, M. Mahdavipour, Roughness in modules, Information Sciences 176 (2006) 3658–3674.[15] B. Davvaz, M. Mahdavipour, Lower and upper approximations in a general approximation space, International Journal of General Systems 37 (2008)

373–386.[16] Z. Gong, B. Sun, D. Chen, Rough set theory for the interval-valued fuzzy information systems, Information Sciences 178 (2008) 1968–1985.[17] T. Iwinski, Algebraic approach to rough sets, Bulletin of the Polish Academy of Sciences – Mathematics 35 (1987) 673–683.[18] O. Kazanci, B. Davvaz, On the structure of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in commutative rings, Information

Sciences 178 (2008) 1343–1354.[19] O. Kazancı, S. Yamak, B. Davvaz, The lower and upper approximations in a quotient hypermodule with respect to fuzzy sets, Information Sciences 178

(2008) 2349–2359.[20] M. Kondo, On the structure of generalized rough sets, Information Sciences 176 (2006) 589600.[21] N. Kuroki, Rough ideals in semigroups, Information Sciences 100 (1997) 139–163.[22] V. Leoreanu-Fotea, B. Davvaz, Roughness in n-ary hypergroups, Information Sciences 178 (2008) 4114–4124.[23] F. Li, Y. Yin, L. Lu, ðh; TÞ-fuzzy rough approximation operators and TL-fuzzy rough ideals on a ring, Information Sciences 177 (2007) 47114726.[24] T.Y. Lin, Topological and fuzzy rough sets, in: R. Slowinski (Ed.), Intelligent Decision Support: Handbook of Applications and Advances of the Rough

Sets Theory, Kluwer Academic Publishers, Boston, 1992, pp. 287–304.[25] J. Liu, Q. Hu, D. Yu, A weighted rough set based method developed for class imbalance learning, Information Sciences 178 (2008) 12351256.[26] Z. Pawlak, A. Skowron, Rudiments of rough sets, Information Sciences 177 (2007) 327.[27] Z. Pawlak, A. Skowron, Rough sets: some extensions, Information Sciences 177 (2007) 2840.[28] Z. Pawlak, A. Skowron, Rough sets and Boolean reasoning, Information Sciences 177 (2007) 4173.[29] Z. Pawlak, Rough sets, International Journal of Information Computer Science 11 (1982) 341–356.[30] D. Pei, On definable concepts of rough set models, Information Sciences 177 (2007) 42304239.[31] J. Pomykala, J.A. Pomykala, The Stone algebra of rough sets, Bulletin of the Polish Academy of Sciences – Mathematics 36 (1988) 495–508.[32] K. Qin, J. Yang, Z. Pei, Generalized rough sets based on reflexive and transitive relations, Information Sciences 178 (2008) 4138–4141.[33] C. Wang, C. Wu, D. Chen, A systematic study on attribute reduction with rough sets based on general binary relations, Information Sciences 178 (2008)

2237–2261.[34] Q.M. Xiao, Z.-L. Zhang, Rough prime ideals and rough fuzzy prime ideals in semigroups, Information Sciences 176 (2006) 725733.[35] X.-P. Yang, Minimization of axiom sets on fuzzy approximation operators, Information Sciences 177 (2007) 38403854.[36] Y. Yang, R. John, Roughness bounds in rough set operations, Information Sciences 176 (2006) 32563267.[37] Y.Y. Yao, Two views of the theory of rough sets in finite universes, International Journal of Approximation Reasoning 15 (1996) 291–317.[38] W. Zhu, Generalized rough sets based on relations, Information Sciences 177 (2007) 49975011.[39] W. Zhu, Topological approaches to covering rough sets, Information Sciences 177 (2007) 14991508.

roughness in mv-algebras

Documents