Studies on Fuzzy representations of fuzzy groups

Abstract

A classical set A is defined as the collection of objects x which belongs to a universal set X. Each member of X can either belong to A or not. We can define the member elements by using the characteristic function _A defined from X to f0; 1g in which 1 represents membership and 0, non-membership. The set defined by a membership function : X ! [0; 1] is called a fuzzy set. The concept of fuzzy set was introduced by Lot A. Zadeh in 1965. Later, Rosenfeld [41] initiated the fuzzification of algebraic structures, by introducing fuzzy groups and discussing some of their properties. The focus of this study is on fuzzy representations. This thesis is organised into five chapters. Definitions and preliminary results from fuzzy set theory, fuzzy operations, fuzzy groups, fuzzy homomorphisms, solvable fuzzy groups which are required in the succeeding chapters are included in chapter 1. After the introduction of fuzzy group by Rosenfeld, fuzzy versions of various algebraic structures were studied by scholars like Abu Osman, Katsars and Liu, Gu Wen-Xiang and so on. Gu developed the concept of M-fuzzy groups. In chapter 2, we analyse the notion of some fuzzy algebraic structures such as order of a fuzzy group, solvable fuzzy group, M-fuzzy group and fuzzy G-modules. The theory of representations has been a powerful analytical tool in the study of groups and group representations. It attempts a classification of homomorphisms of abstract _nite group into groups of matrices or linear transformations. For an indepth study in representation theory, module theoretic approach is more suited and it gives more elegance to the theory. So, in the study of representations, G-module structure is widely used. The representation theory was developed on the basis of embedding a group G into a general linear group GL(V ).