3 Nilpotent, commutator, and solvable fuzzy subgroups

Abstract

As the title suggests, this chapter is concerned with nilpotent, commutator, and solvable fuzzy subgroups. We give two approaches for the development of these notions, namely one via an ascending chain of fuzzy subgroups and one via descending chain of fuzzy subgroups. The results of this chapter are mainly from [5, 8, 18]. 3.1 Commutative Fuzzy Subsets and Nilpotent Fuzzy Subgroups In this section, we introduce the notion of a commutative fuzzy subset and the notion of a nilpotent fuzzy subgroup. We show that the nilpotence of a group can be completely characterized by the nilpotence of its fuzzy subgroups. For the relevant classical group-theoretic results and definitions, the reader is referred to [6, 17]. Let µ be a fuzzy subset of a groupoid G. The normalizer N (µ) of µ in G is the set {x ∈ G | µ(xy) = µ(yx) for all y ∈ G} and µ is called normal in G if N (µ) = G. If G is a group, then µ(e) is called the tip of µ. Definition 3.1.1. Let µ be a fuzzy subset of a semigroup G. Let Z(µ) = {x ∈ G | µ(xy) = µ(yx) and µ(xyz) = µ(yxz) for all y, z ∈ G}. Then µ is called commutative in G if Z(µ) = G. We adopt the terminology from [8] and call Z(µ) in Definition 3.1.1 the cen-tralizer of µ in G. If G has a right identity, then the equality µ(xy) = µ(yx) in Definition 3.1.1 is redundant. If G is a semigroup, we let Z(G) denote the center of G. It is clear that Z(G) ⊆ Z(µ) ⊆ N (µ) and that Z(G) = Z(µ) = N (µ) is possible. In the following example, recall that the notation a, b | a 3 = e = b 2 , ba = a 2 b denotes the group generated by a and b, where a and b satisfy the properties a 3 = e = b 2 and ba = a 2 b.