(m,n)-Semihyperrings and an Algebra of Fuzzy (m,n)-Semihyperrings

TL;DR

This paper introduces a new algebraic structure called $(m,n)$-semihyperring, generalizing semihyperrings, and explores their properties, homomorphisms, and fuzzy set extensions, providing foundational theorems and relationships.

Contribution

It defines $(m,n)$-semihyperrings, develops their basic properties, and extends the concept to fuzzy sets, establishing new theorems and relationships in this algebraic framework.

Findings

Defined $(m,n)$-semihyperring and basic properties

Proved theorems on homomorphisms and quotient structures

Extended to fuzzy $(m,n)$-semihyperrings and established relationships

Abstract

We propose a new class of algebraic structure named as \emph{$(m,n)$-semihyperring} which is a generalization of usual \emph{semihyperring}. We define the basic properties of $(m,n)$-semihyperring like identity elements, weak distributive $(m,n)$-semihyperring, zero sum free, additively idempotent, hyperideals, homomorphism, inclusion homomorphism, congruence relation, quotient $(m,n)$-semihyperring etc. We propose some lemmas and theorems on homomorphism, congruence relation, quotient $(m,n)$-semihyperring, etc and prove these theorems. We further extend it to introduce the relationship between fuzzy sets and $(m,n)$-semihyperrings and propose fuzzy hyperideals and homomorphism theorems on fuzzy $(m,n)$-semihyperrings and the relationship between fuzzy $(m,n)$-semihyperrings and the usual $(m,n)$-semihyperrings.