SOME PROPERTIES OF T-FUZZY GENERALIZED SUBGROUPS

In this paper, we deal with Molaei’s generalized groups. We de- fine the notion of a fuzzy generalized subgroup with respect to a t-norm (or T-fuzzy generalized subgroup) and give some related properties. Especially, we state and prove the Representation Theorem for these fuzzy generalized subgroups. Next, using the concept of continuity of t-norms we obtain a cor- respondence between TF(G), the set of all T-fuzzy generalized subgroups of a generalized group G, and the set of all T-fuzzy generalized subgroups of the corresponding quotient generalized group. Subsequently, we study the quo- tient structure of T-fuzzy generalized subgroups: we define the notion of a T-fuzzy normal generalized subgroup, give some related properties, construct the quotient generalized group, state and prove the homomorphism theorem. Finally, we study the lattice of T-fuzzy generalized subgroups and prove that TF(G) is a Heyting algebra.