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 semihyper‌rings 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.