Membership Functions From Similarity Relations

Abstract

Let G be a group and μ a fuzzy subgroup of G. Then μ can be thought to be the membership function of a fuzzy subgroup of G. In this chapter, we sometimes refer to μ in this way. We show that μ satisfies the equation μ(x) = σ(e,x), where σ is a similarity relation on G which is invariant under left-translation. We also show that under certain natural assumptions the elements x of G can be represented as permutations P^x of a suitable universe Ω such that μ(x) equals the proportion of elements in Ω which are fixed by P^x. These results provide a deeper insight on the relationship of the group operation to the membership values μ(x).