Fuzzy Subgroups and Digraphs Induced by Fuzzy Subgroups
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Paul J. Allen
1
(Department of Mathematics, University of Alabama, Tuscaloosa, U.S.A..)
Joseph Neggers
2
(Department of Mathematics, University of Alabama, Tuscaloosa, U.S.A..)
Hee Sik Kim
3
(Department of Mathematics, Hanyang University, Seoul, Korea.)
کلید واژه: Fuzzy subgroup, $\mu$-product relation, Fuzzy normal, Digraph, $(\mu, \nu)$-homomorphism.,
چکیده مقاله :
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 $G$.
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 $G$.
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