Fuzzy subgroups: Some characterizations

Abstract

The notion of a fuzzy set was first introduced in 1965 by Zadeh [12]. Since then the theory of fuzzy sets has gone through remarkably rapid strides with well over 4000 papers by now (see, for example, the bibliography [3]), several textbooks (for example, [2 and 6]), and an international journal solely devoted to it [S]. The theory has found wideranging applications in such diverse fields as automata, control theory, decision theory, and social behavior pattern studies, to name just a few. For an interesting, expository account see Rosenfeld [lo], and also the book review [7]. In [9], Rosenfeld introduced the notion of a fuzzy group and showed that many concepts of group theory can be extended in an elementary manner to develop the theory of fuzzy groups. In particular, he characterized all the fuzzy subgroups of cyclic groups of prime order [9, Proposition 5.101. Fuzzy groups were further investigated by Das in [1] where he characterized all the fuzzy subgroups of finite cyclic groups [l, Theorem 4.21. In this connection, he introduced the notion of “level subgroups” of a fuzzy subgroup which is based on the notion of a “level subset” introduced earlier by Zadeh. For a finite group Das showed that these level subgroups of a fuzzy subgroup form a chain [ 1, Corollary 3.11 and also constructed a fuzzy subgroup whose chain of level subgroups is maximal in the following special cases when the group is: (i) a direct product of cyclic groups of prime orders, (ii) abelian [l, Theorem 3.41. We analyze the level subgroups of a fuzzy subgroup in more detail and investigate whether the family of level subgroups of a fuzzy subgroup deter-