‎Fuzzy Subgroups and Digraphs Induced by Fuzzy Subgroups

Abstract

Given a fuzzy subgroup $\mu$ of a group $G$‎, ‎$x\rhd_uy$ if and only if $\mu(xy) < \mu(yx)$ defines a directed relation with an associated digraph $(G‎, ‎\rhd_u)$‎. ‎We consider $(\mu‎, ‎\nu)$-homomorphisms $\varphi‎: ‎(G‎, ‎\mu)\to (H‎, ‎\nu)$ where $\mu$ and $\nu$ are fuzzy subgroups of $G$ and $H$ respectively and the preservation of properties of the digraphs $(G‎, ‎\rhd_u)$ several of which are also noted here‎, ‎e.g.‎, ‎$(G‎, ‎\rhd_u)$ is an anti-chain if and only if $\mu$ is a fuzzy normal subgroup of the group $G$‎.

Keywords

• fuzzy‎ ‎subgroup‎
• ‎$mu$-product relation‎
• fuzzy normal‎
• digraph‎
• ‎$(mu‎
• ‎nu)$-homomorphism‎‏