On exact category of (m, n)-ary hypermodules

Najmeh Jafarzadeh,Reza Ameri

doi:10.29252/cgasa.12.1.69

Abstract

We introduce and study category of (m, n)-ary hypermodules as a generalization of the category of (m, n)-modules as well as the category of classical modules. Also, we study various kinds of morphisms. Especially, we characterize monomorphisms and epimorphisms in this category. We will proceed to study the fundamental relation on (m, n)-hypermodules, as an important tool in the study of algebraic hyperstructures and prove that this relation is really functorial, that is, we introduce the fundamental functor from the category of (m, n)-hypermodules to the category (m, n)-modules and prove that it preserves monomorphisms. Finally, we prove that the category of (m, n)-hypermodules is an exact category, and, hence, it generalizes the classical case.

Highlights

Introduction and preliminariesThe concept of a hypergroup was introduced by Marty in [19]
Recall from [18] that normal and conormal categories with kernels and cokernels are exact if every morphism α : A → B can be written as a composition A → I → B where q is an epimorphism and ν is a monomorphism
We constructed the category of (m, n)−hypermodules and proved that it is an exact category in the sense that it is normal and conormal with kernels and cokernels in which every morphism α has a factorization α = νq, where q is an epimorphism and ν is a monomorphism

Summary

Introduction and preliminaries

The concept of a hypergroup was introduced by Marty in [19]. Afterwards, because of many applications of this theory in both pure and applied sci-. The objects of R(m,n) − KHmod are the canonical (m, n)-hypermodules over Krasner (m, n)-hyperring and all morphisms are multivalued homomorphisms. Let (A, f1, g1), (B, f2, g2) be (m, n)-hypermodules over the (m, n)hyperring R and φ : A → B be a monomorphism, we prove φ∗ : A/ ∗ → B/ ∗ is monic. For any [γ, C] ∈ Q] knowing that γ is a homomorphism which is onto (that is, an epimorphism in h(m,n) − Rmod; by Theorem 2.3), if we set B = kerγ, in view of Theorem 4.1, there exists an isomorphism α : A/B → C such that α ◦ πB = γ ( α is an isomorphism because α is onto and kerγ = B = kerπB) This means that [γ, C] = [πB, A/B] = η(A/B), which completes the bijective of η.

GA iC f

Conclusions and future works