A CATEGORICAL CONNECTION BETWEEN CATEGORIES (m, n)-HYPERRINGS AND (m, n)-RING VIA THE FUNDAMENTAL RELATION Γ

Abstract

Let R be an (m,n)-hyperring. The Γ∗-relation on R in the sense of Mirvakili and Davvaz [?] is the smallest strong compatible relation such that the quotient R∕Γ∗ is an (m,n)-ring. We use Γ∗-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∕Γ∗ 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.