Fuzzy Normal Subgroups in Fuzzy Subgroups

The theory of fuzzy sets was inspired by Zadeh [10]. Subsequently, Rosenfeld introduced the concept of a fuzzy subgroup of a group [9]. Fuzzy cosets and fuzzy normal subgroups of a group G have been studied in [3, 5, 8]. In [4], the ring of cosets of a fuzzy ideal was constructed. Let A and B be fuzzy subgroups of G such that B ⊆ A. The purpose of this paper is to introduce the notion of fuzzy cosets and fuzzy normality of B in A. These ideas differ from those in [3, 5, 8] since there A = δG , the characteristic function of G. If B is fuzzy normal in A, then the set of all fuzzy cosets of B in A forms a semigroup under a suitable operation. Structure properties of A/B and A are determined. Throughout this paper G denotes a group and L denotes a completely distributive lattice. A fuzzy subgroup A of G is a fuzzy subset of G (a function of G into L) such that ∀x, y ∈ G, A(xy−1) ≥ inf{A(x), A(y)}. We let e denote the identity of G and 0, 1 the least element, the greatest element of L respectively. If X and Y are fuzzy subsets of G, we say that X ⊆ Y if and only if ∀x ∈ G, X (x) ≤ Y (x). For any x ∈ G, t ∈ L , we let xt denote the fuzzy subset of G defined by ∀y ∈ G, xt(y) = 0 if y 6= x and xt(y) = t if y = x . We call xt a fuzzy singleton. B and A always denote fuzzy subgroups of G such that B ⊆ A. If t ∈ L , we let Bt = {x ∈ G | B(x) ≥ t}. It follows easily that if t ∈ Im(B), then Bt is a subgroup of G. Bt is called a level subgroup of G [1]. We let B∗ = BB(e). For any fuzzy subgroup A of G we assume that A(e) > 0. N denotes the set of positive integers.