Fuzzy congruences on groups
by
Michiro Kondo
Tokyo Denki University

In the theory of groups, there exists a close relationship between normal subgroups and congruences. It is a natural question to extend the relationships of these to the case of fuzzy group theory. We here define fuzzy congruences on groups and fuzzy quotient groups by fuzzy congruences and investigate their properties. Rosenfeld defined fuzzy subgroupoids and proved that a homomorphic image of a fuzzy subgroupoid with the sup property was a fuzzy subgroupoid, and hence that a homomorphic image of a fuzzy subgroup with the sup property was a fuzzy subgroup. This theorem needs the sup property. But we can show the theorem without sup property, that is, a homomorphic image of a fuzzy subgroup is a fuzzy subgroup. Moreover, Mukherjee and Bhattacharya showed that if [`A] is a fuzzy subgroup of a fintite group G such that all the level subgroups of G are normal subgroups then [`A] is a fuzzy normal subgroup. We can also prove the theorem without finiteness using the transfer principle which is a fundamental tool we have developped here.

We show that

1. The lattice FNS(G) of all fuzzy normal subgroups of a group G is isomorphic to the lattice FCon(G) of all fuzzy congruences on G ;

2. FNS_a(G) forms a modular lattice for every a ∈ [0, 1] ;

3. Let G and G' be groups and f:G → G' be a homomorphism. If [`A] is a fuzzy (normal) subgroup of G then f[[`A]] is a fuzzy (normal) subgroup of G' ;

4. Let G, G' and f be as above. If [`A] is a fuzzy normal subgroup of G' then G/f<sup>-1</sup>([`A]) ≅ f(G)/[`A].

In this talk, I will show that these results are obtained by the crisp group theory with so-called transfer principle. This means that the transfer principle is a very important tool to investigate the fuzzy theory.

Date received: March 31, 2008