On Commutators of Fuzzy Multigroups

Abstract

Fuzzy multigroup is an application of fuzzy multiset to group theory. Although, a lots have been done on the theory of fuzzy multigroups, some group's theoretic notions could still be investigated in fuzzy multigroup context. Certainly, the idea of commutator is one of such group's theoretic notions yet to be studied in the environment of fuzzy multigroups. Hence, the aim of this article is to establish the notion of commutator in fuzzy multigroup setting. A number of some related results are obtained and characterized. Among several results that are obtained, it is established that, if $A$ and $B$ are fuzzy submultigroups of a fuzzy multigroup $C$, then $[A, B]\subseteq A\cup B$ holds. Some homomorphic properties of commutator in fuzzy multigroup context are discussed. The notion of admissible fuzzy submultisets $A$ and $B$ of $C\in FMG(X)$ under an operator domain $\mathcal{D}$ is explicated, and it is shown that $(A,B)$ and $[A,B]$ are $\mathcal{D}$-admissible.