Permutable indistinguishability operators, perfect vague groups and fuzzy subgroups

Permutability between T-indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures. It will be shown that the sup −T product E ∘ F of two T-indistinguishability operators is also a T-indistinguishability operator if and only if E and F are permutable T-indistinguishability operators (i.e., E ∘ F = F ∘ E). This property will be related to the study of fuzzy subgroups, fuzzy normal subgroups and vague groups. The aggregation of fuzzy subgroups will also be analyzed.