a strong regular relation on ?-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 ?-semihyperrings is a generalization of semirings, a generalization of semihyper rings and a generalization of ?-semirings. in this paper, we introduce an equivalence relation ?* on a ?-semihyperrings r and we show that it is strongly regular. furthermore, r/?*, the set of all equivalence classes of this relation is a ?/?*-semiring. the relation ?* is called the fundamental relation and the ?-semiring r/?* is called the fundamental semiring. fundamental relations are the main tools in the study of ?-semihyperrings. we present some results about fundamental relations and fundamental semirings. finally, we show that there is a covariant functor between the category of ?-semihyperrings and the category of semirings.