Some algebraic applications of soft sets

Abstract

The paper presents an algebraic construction of the basic notions of soft set theory. Here we introduce the definition of bijective soft groups, dependent and independent soft groups. Some of their properties are investigated. Also we investigate the relationships between bijective soft groups and classical groups. In addition, we express some applications of bijective soft groups on coding theory.

Introduction

The original concept of soft set was firstly introduced by Moldtsov [1]. His pioneer paper has undergone tremendous growth and applications in the last few years. Maji et al. [2] give an application of soft set theory in a decision making problem by using the rough sets and they conducted a theoretical study on soft sets in a detailed way [3]. Chen et al. [4] proposed a reasonable definition of parameterizations reduction of soft sets and compared them with the concept of attributes reduction in rough set theory.

The algebraic structures of set theories which deal with uncertainties have been studied by some authors. Rosenfeld [5] proposed fuzzy groups to establish results for the algebraic structures of fuzzy sets. Rough groups were defined by Biswas et al. [6] and some others (i.e. Bonikowaski [7], Iwinski [8]) studied algebraic properties of rough sets.

Many authors studied on applications of soft sets. Çağman et al.[9] defined products of soft sets and uni-int decision function and then using this definition, they constructed an uni-int decision making method which selects a set of optimum elements from the alternatives. Babitha et al. [10] introduced a sub soft set of the Cartesian product of the soft set and many related concepts. Xiau et al. [11] proposed the notion of exclusive disjunctive soft sets and studied some of its operations. Gong et al. [12] defined the concept of bijective soft set and some of its operations and studied its several properties. Feng et al. [13] established an interesting connection between two mathematical approaches to vagueness: rough sets and soft sets. They also defined new types of soft sets such as full soft sets, intersection complete soft sets and partition soft sets.

In [14], Aktaş and Çağman introduced the notion of a soft group and showed that many concepts of group theory can be extended in an elementary manner to develop the theory of soft groups. Since then, many author have developed soft algebraic concepts. Jun et al. [16] studied soft ideals and idealistic soft BCK/BCI-algebras. Acar et al. [17] introduced initial concepts of soft rings. Aygunoğlu et al. [18] introduced the concept of fuzzy soft group and, in the meantime, they studied its properties and structural characteristics. Atagun and Sezgin [19] introduced and studied the concepts of soft sub-rings, soft ideal of a ring and soft sub fields of a field.

In this paper, by using Gong's [12] definition of the bijective soft set and Aktaş's definition of the soft group, we introduce definition of bijective soft groups and show its basic properties with a theoretical study. The presentation of the rest of the paper is organized as follows. In Section 2, the fundamental properties of soft sets, bijective soft set and soft groups are presented. In Section 3, the concepts of bijective soft group and bijective soft subgroup are defined and their basic properties are studied and mentioned some applications of bijective soft groups on coding theory. In Section 4, using the concept of bijective soft group, we obtain some basic results of classical groups. Finally, in Section 5, conclude the present paper with suggestions for future work.