Fundamental relation on fuzzy hypermodules

Abstract

The fundamental relation $\theta ^{*}$ on M can be defined as the smallest equivalence relation such that the quotient $M/\theta ^{*}$ be a module over the corresponding fundamental ring such that $M/\theta ^{*}$ as a group is not abelian. Moreover, the fundamental ring is not commutative with respect to both sum and product. Now, we would like the fundamental module as a group to be abelian and the fundamental ring to be commutative with respect to both sum and product. Also, we assign fundamental functor between the category of fuzzy hypermodules and the category of abelian groups to convey its features and related commutative diagram. Finally, we find necessary and sufficient conditions such that $\theta$ is transitive.