Abstract
Using the notions of belonging (is an element of) and quasi-k-coincidence (q(k)) of a fuzzy point with a fuzzy set, we define the concepts of ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar)-fuzzy normal subgroups and ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar)-fuzzy cosets which is a generalization of fuzzy normal subgroups, fuzzy coset, ((is an element of) over bar, (is an element of) over bar boolean OR (q) over bar)-fuzzy normal subgroups and ((is an element of) over bar, (is an element of) over bar boolean OR (q) over bar)-fuzzy cosets. We give characterizations of an ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar)-fuzzy normal subgroup and ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar)-fuzzy coset, and deal with several related properties. The important achievement of the study with an ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar)-fuzzy normal subgroup and ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar)-fuzzy cosets is the generalization of that the notions of fuzzy normal subgroups, fuzzy coset, ((is an element of) over bar, (is an element of) over bar boolean OR (q) over bar)-fuzzy normal subgroups and ((is an element of) over bar, (is an element of) over bar boolean OR (q) over bar)-fuzzy cosets. We prove that the set of all ((is an element of) over bar, (is an element of) over bar boolean OR (q(k)) over bar )fuzzy cosets of G is a group, where the multiplication is defined by (f) over left arrow (x) . (f) over left arrow (y) = (f) over left arrow (xy) for all x, y is an element of G. If (f) over tilde : F -> [0, 1] is defined by (f) over tilde((f) over left arrow (x)) = f (x) for all x is an element of G. Then (f) over tilde is a fuzzy normal subgroup of F.