A Strong Regular Relation on Γ-Semihyperrings

The concept of algebraic hyperstructures introduced by marty as a generalization of ordinary algebraic structures. in an ordinary algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. the concept of γ semihyperrings is a generalization of semirings, a generalization of semihyper- rings and a generalization of γsemirings. in this paper, we introduce an equivalence relation γ* on a γ-semihyperrings r and we show that it is strongly regular. furthermore, r/γ*, the set of all equivalence classes of this relation is a γ/β*-semiring. the relation γ* is called the fundamental relation and the γ-semiring r/γ* is called the fundamental semiring. fundamental relations are the main tools in the study of γ-semihyperrings. we present some results about fundamental relations and fundamental semirings. finally, we show that there is a covariant functor between the category of γ-semihyperrings and the category of semirings.