Some Properties of $T$-fuzzy Generalized Subgroups

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In this paper, we deal with Molaeiʹs generalized groups. We define 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 correspondence 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 quotient 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.