A Categorical Connection Between Categories $(m,n)$-Hyperrings and $(m,n)$-Ring via the Fundamental Relation $\Gamma^*$

Ameneh Asadi, Reza Ameri, Morteza Norouzi

Let $R$ be an $(m,n)$-hyperring. The $\Gamma^\ast$-relation on $R$ in the sense of Mirvakili and Davvaz \cite{[26]} is the smallest strong compatible relation such that the quotient $R/\Gamma^\ast$ is an $(m,n)$-ring. We use $\Gamma^\ast$-relation to define a fundamental functor, $F$ from the category of $(m,n)$-hyperrings to the category of $(m,n)$-rings. Also, the concept of a fundamental $(m,n)$-ring is introduced and it is shown that every $(m,n)$-ring is isomorphic to $R/\Gamma^\ast$ for a nontrivial $(m,n)$-hyperring $R$. Moreover, the notions of partitionable and quotientable are introduced and their mutual relationship is investigated. A functor $G$ from the category of classical $(m,n)$-rings to the category of $(m,n)$-hyperrings is defined and a natural transformation between the functors $F$ and $G$ is given.