Generalized lower and upper approximations in a ring

Abstract

In this paper, the concepts of set-valued homomorphism and strong set-valued homomorphism of a ring are introduced, and related properties are investigated. The notions of generalized lower and upper approximation operators, constructed by means of a set-valued mapping, which is a generalization of the notion of lower and upper approximation of a ring, are provided. We also propose the notion of generalized lower and upper approximations with respect to an ideal of a ring which is an extended notation of rough ideal introduced lately by Davvaz [B. Davvaz, Roughness in rings, Information Science 164 (2004) 147–163] in a ring and discuss some significant properties of them.

Introduction

Rough set theory [39], [40], [41], a new mathematical approach to deal with inexact, uncertain or vague knowledge, has recently received wide attention on the research areas in both of the real-life applications and the theory itself. It has found practical applications in many areas such as knowledge discovery, machine learning, data analysis, approximate classification, conflict analysis, and so on, see [20], [27], [32], [35], [42], [44], [45], [58], [61], [62]. In the theory of rough set [39], it is important to construct a pair of upper and lower approximation operators based on available information. The Pawlak approximation operators are defined by an equivalence relation. However, the requirement of an equivalence relation in Pawlak rough set models seem to be a very restrictive condition that may limit the applications of the rough set models. Thus one of the main directions of research in rough set theory is naturally the generalization of the Pawlak rough set approximations. For instance, the notations of approximations are extended to general binary relations, coverings, completely distributive lattices, fuzzy lattices, and Boolean algebras [2], [10], [19], [42], [48], [51], [55], [59], [60], [63], [66], [67], [68], [69].

Some researches studied algebraic properties of rough sets. Biswas and Nanda [1] applied the notion of rough sets to algebra and introduced the notion of rough subgroups. Kuroki [22], introduced the notion of a rough ideal in a semigroup. Kuroki and Wang [21] gave some properties of the lower and upper approximations with respect to the normal subgroups. In addition, Kuroki and Mordeson [23] studied the structure of rough sets and rough groups. Mordeson [36] used covers of the universal set to define approximation operators on the power set of the given set. In Refs. [6], [7], Davvaz concerned a relationship between rough sets and ring theory and considered a ring as a universal set and introduced the notion of rough ideals and rough subrings with respect to an ideal of a ring. In Ref. [17], Kazancı 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 [8], also see [4], [5], [9], [12], [16], [18], [31], [47], [50], [54], [56]. Dubois and Prade [13] introduced the problem of fuzzification of a rough set. Many authors analyzed the concept of a fuzzy rough set. For example, see [7], [14], [30], [34], [38], [57], [64]. The study of roughness in n-ary hypergroups introduced in [25], is utilized in [24] in order to introduce fuzzy rough n-ary subhypergroups.

The initiation and majority of studies on rough sets for algebraic structures such as semigroups, groups, rings, and modules have been concentrated on a congruence relation. The congruence relation, however, seems to be restrict the application of the generalized rough set model for algebraic sets. To solve this problem, Davvaz [11] introduce the concept of a set-valued homomorphism for groups. From this point of view, in Section 2, the concept of generalized rough approximation operators on rings is introduced and its basic properties are discussed. In Section 3, we define (strong) set-valued homomorphism and its properties are studied. Then we explore some fundamental properties of generalized lower and upper approximation operators constructed by means of a set-valued mapping in a ring. Specially, we introduce the concept of uniform set-valued homomorphism and prove that every set-valued homomorphism is uniform. Also, we introduce the notion of generalized rough subring (resp. ideal) with a set-valued homomorphism. Finally, in Section 4, the concept of the generalized lower and upper approximation operators, constructed by means of a set-valued homomorphism, with respect to an ideal of a ring is presented and we have examined some properties of the lower and upper approximation operators in a ring.