Fuzzy Subgroups and Digraphs Induced by Fuzzy Subgroups
محورهای موضوعی : Transactions on Fuzzy Sets and Systems
Paul J. Allen
1
,
Joseph Neggers
2
,
Hee Sik Kim
3
1 - Department of Mathematics, University of Alabama, Tuscaloosa, U.S.A..
2 - Department of Mathematics, University of Alabama, Tuscaloosa, U.S.A..
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 $G$.
[1] M. Akgul, Some properties of fuzzy groups, J. Math. Anal. Appl., 133 (1988), 93-100.
[2] P. S. Das, Fuzzy groups and level subgroups, J. Math. Anal. Appl., 84 (1981), 264-269.
[3] R. Kumar, Fuzzy algebra, Vol 1, University of Delhi, Delhi, (1993).
[4] W. J. Liu , Fuzzy invariant subgroups and fuzzy ideals, Fuzzy Sets and Systems, 8 (1982), 133-139.
[5] J. N. Mordeson, K. R. Bhutani and A. Rosenfeld, Fuzzy group theory, Springer, New York, (2005).
[6] J. N. Mordeson and D. S. Malik, Fuzzy commutative algebras, World Scienti c, Singapore, (1998).
[7] N. P. Mukherjee and P. Bhattacharya, Fuzzy normal subgroups and fuzzy cosets, Information Sciences, 34 (1984), 225-239.
[8] J. Neggers and H. S. Kim, Modular semigroups and posets, Semigroup Forum, 53 (1996), 57-62.
[9] J. Neggers and H. S. Kim, Algebras associated with posets, Demonstratio Math., 34 (2001), 13-23.
[10] J. Neggers and H. S. Kim, Fuzzy posets on sets, Fuzzy Sets and Systems, 117 (2001), 391-402.
[11] J. Neggers and H. S. Kim, Fuzzy pogroupoids, Inform. Sci., 175 (2005), 108-119.
[12] A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl., 35 (1971), 512-517.
