Roughness in MV-algebras

Abstract

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

Introduction

The theory of rough sets was introduced by Pawlak [29] in 1982, also see [26], [27], [28]. The theory of rough sets is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. A key concept in Pawlak rough set model is an equivalence relation. The equivalence classes are the building blocks for the construction of the lower and upper approximations. The lower approximation of a given set is the union of all the equivalence classes which are subsets of the set, and the upper approximation is the union of all the equivalence classes which have a non-empty intersection with the set. The objects of the given universe U can be divided into three classes with respect to any subset $A\subseteq U$:

• the objects, which are definitely in A;

• the objects, which are definitely not in A;

• 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 upper approximation. Some authors, for example, Iwinski [17], and Pomykala and Pomykala [31] have studied algebraic properties of rough sets. The lattice theoretical approach has been suggested by Iwinski [17]. Pomykala and Pomykala [31] showed that the set of rough sets forms a Stone algebra. Comer [7] presented an interesting discussion of rough sets and various algebras related to the study of algebraic logic, such as Stone algebras and relation algebras. A natural question is what will happen if we substitute an algebraic system instead of the universe set. Biswas and Nanda [1] introduced the notion of rough subgroups. Kuroki in [21], introduced the notion of a rough ideal in a semigroup, also see [34]. Davvaz in [8] introduced the notion of rough subrings (respectively ideal) with respect to an ideal of a ring, also see [11]. In [18], Kazanci and Davvaz introduced the notions of rough prime (primary) ideals and rough fuzzy prime (primary) ideals in a ring and gave some properties of such ideals. Rough modules have been investigated by Davvaz and Mahdavipour [14]. In [22], Leoreanu-Fotea and Davvaz introduced the concept of roughness in n-ary hypergroups, also see [19]. For more information about rough sets and algebraic 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 infinite valued $Ł$ukasiewicz propositional calculus. An MV-algebra A is an abelian monoid $〈A\text{,}0\text{,}\oplus 〉$ equipped with an operation $\ast$ such that $(x^{\ast }{)}^{\ast }=x\text{,}x\oplus 0^{\ast }=0^{\ast }$ and finally, $(x^{\ast }\oplus y{)}^{\ast }\oplus y=(y^{\ast }\oplus x{)}^{\ast }\oplus x$. An example of an MV-algebra is given by the real unit interval [0, 1] equipped with the operations $x^{\ast }=1-x$ and $x\oplus y=\min (1\text{,}x+y)$. Valid equations yield new valid equations by substituting equals for equals. Chang’s Completeness theorem states that in this way one obtains from the above equations every valid equation in the MV-algebra [0, 1]. Boolean algebra stand to Boolean logic as MV-algebra stand to $Ł$ukasiewicz infinite 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 respect to an ideal of an MV-algebra, which is an extended notion of an ideal in an MV-algebra. We give some properties of the lower and the upper approximations in an MV-algebra.