Abstract
We propose a new class of algebraic structure named as (m, n)-semihyperring which is a generalization of usual 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)-semihyper-rings.
Highlights
A semihyperring is essentially a semiring in which addition is a hyperoperation [1]
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
This section deals with the definitions of hyperideals, homomorphism, congruence relation, quotient of (m, n)-semihyperring and the theorems based on these definitions
Summary
A semihyperring is essentially a semiring in which addition is a hyperoperation [1]. Semihyperring is in active research for a long time. Zadeh [6] introduced the notion of a fuzzy set that is used to formulate some of the basic concepts of algebra. Fuzzy hyperideals of semihyperrings are studied by [1,10,11]. In [13] Davvaz studied the fuzzy hyperideals of the Krasner (m, n)-hyperring. We introduce the notion of the generalization of usual semihyperring and called it as (m, n)semihyperring and set fourth some of its properties, we introduce fuzzy (m, n)-semihyperring and its basic properties and the relation between fuzzy (m, n)-semihyperring and its associated (m, n)-semihyperring. This section deals with the definitions of hyperideals, homomorphism, congruence relation, quotient of (m, n)-semihyperring and the theorems based on these definitions.
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