A new attitude coupled with fuzzy thinking to fuzzy group and subgroup

Abstract

In this paper, we shall embark on the study of the algebraic object known as a fuzzy group which serves as one of the fundamental building blocks for the subject which is called fuzzy abstract algebra. In our opinion, the fuzzy algebraic systems are usually sets on whose elements we can operate algebraically by this we mean that we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set and, in addition, we assume that these fuzzy algebraic operations are subject to certain rules, which are explicitly spelled out in what we call the axioms or postulates defining the system. In this abstract setting we then attempt to prove theorems about these very general structures. We should like to stress that these fuzzy algebraic systems and their axioms, must come from the experience of looking at many examples. Namely, they should be rich in meaningful results. Hence, the acceptable definition of fuzzy group and subgroup are presented with binary operations and on the basis of the specified parameter, called ambiguity rank, which fulfils the basic requirements. The properties of these fuzzy groups and their fundamental qualities are discussed and then, the several illustrative examples were given. The future prospect of this paper is a new attitude to fuzzy basic mathematics, which will be referred to in the end.