Direct Products of Fuzzy Subgroups and Fuzzy Cyclic Subgroups

Abstract

In Chapter 6, a necessary and sufficient condition for a fuzzy subgroup to be a weak direct sum of fuzzy subgroups was obtained by employing known structure theorems for Abelian groups. In [31] and [23], the problem of expressing a normal fuzzy subgroup of a direct product of groups as a fuzzy direct product of certain subgroups of a group G was examined. This is essential in order to obtain structure theorems involving fuzzy subgroups of a group G and fuzzy subgroups of it subgroups. In this chapter, we first we introduce the notion of the fuzzy direct product of fuzzy subgroups of subgroups of a group and employ these ideas to the problem in group theory of obtaining conditions under which a group G can be expressed as the direct product of its normal subgroups. We extend the definition of weak direct sum introduced in Chapter 6 for Abelian groups to the general case and establish a one-to-one correspondence between these two fuzzy direct products. Thus the relevant results of fuzzy direct products can be applied to fuzzy weak direct products.