A Note on the Lattice of Fuzzy Subgroups of a Finite Group
The main goal of this note is to discuss the possibility that the set FL(G) of all fuzzy subgroups of a finite group G to be fully ordered. We prove that FL(G) is not fully ordered under the usual fuzzy set inclusion, for any nontrivial finite group G.We also develop a method to endow FL(G) with a fully ordering relation. In particular, the same problems are treated for the set FN(G) of all fuzzy normal subgroups of G.
Keywords: fuzzy subgroup lattices, fuzzy normal subgroup lattices, fully ordering relations, equivalence relations.
MSC (2010): Primary 20N25, 03E72; Secondary 20D30, 20E15