SOME PROPERTIES OF T-FUZZY GENERALIZED SUBGROUPS

In this paper, we deal with Molaei’s generalized groups. We definethe notion of a fuzzy generalized subgroup with respect to a t-norm (orT-fuzzy generalized subgroup) and give some related properties. Especially,we state and prove the Representation Theorem for these fuzzy generalizedsubgroups. Next, using the concept of continuity of t-norms we obtain a correspondencebetween TF(G), the set of all T-fuzzy generalized subgroups of ageneralized group G, and the set of all T-fuzzy generalized subgroups of thecorresponding quotient generalized group. Subsequently, we study the quotientstructure of T-fuzzy generalized subgroups: we define the notion of aT-fuzzy normal generalized subgroup, give some related properties, constructthe quotient generalized group, state and prove the homomorphism theorem.Finally, we study the lattice of T-fuzzy generalized subgroups and prove thatTF(G) is a Heyting algebra.