The notion of ternary $$\Gamma $$ -semihyperrings is a generalization of semirings. In a ternary $$\Gamma $$ -semihyperring, addition is a hyperoperation and multiplication is a ternary hyperoperation. Our main purpose of this paper is to introduce regular and simple ternary $$\Gamma $$ -semihyperrings and study the notions of right (left) fundamental semirings, and we show that there is a covariant functor between the category of relocation ternary $$\Gamma $$ -semihyperrings and semirings.

The concept of a \(\Gamma \)-semihyperring is a generalization of a semiring, a generalization of a semihyperring and a generalization of a \(\Gamma \)-semiring. In this paper, we introduce the notion of \(\Gamma \)-semihyperring, then we prove some results in this respect and present several examples of \(\Gamma \)-semihyperrings. The notions of ideal and homomorphism are introduced and some properties of them are presented. Moreover, we study some properties of regular and strongly regular relations on a \(\Gamma \)-semihypergroup.

In this paper, we generalize the notion of algebra over a field. A Γ-algebra is an algebraic structure consisting of a vector space V , a groupoid Γ together with a map from V ×Γ×V to V , usually called multiplication. We introduce the notion of Γdimension and give some examples and prove some properties of Γ-algebras. Then, we give some results about m× n real matrices. Also, we study the notion of regular Γ-algebra and we obtain some results in this respect. Finally, we define the notions of T -functor and H-system over a Γ-algebra and prove some results. Moreover, we see that there exists a covariant functor between the categories of Γalgebras and algebras. We see that this functor is exact.

In this paper, we generalize the notion; of algebra over afield. A Gamma-algebra is an algebraic structure consisting of a vector space V, a groupoid Gamma together with a map from V x Gamma x V to V, usually called multiplication. We introduce the notion of Gamma- dimension and give some examples and prove some properties of Gamma-algebras. Then, we give some results about m x n real matrices. Also, we study the notion of regular Gamma-algebra and we obtain some results in this respect. Finally, we define the notions of T-functor and H-system over a Gamma-algebra and prove some results. Moreover, we see that there exists a covariant functor between the categories of Gamma- algebras and algebras. We see that this functor is exact.

The notion of rough sets was introduced by Z. Pawlak in 1982. The concept of Gamma-semihyperring is a generalization of semihyperring, Gamma-semiring and semiring. In this paper, we study the notion of a rough (rough prime) ideal in a Gamma-semihyperring. Also, we discuss the relation between the upper and lower rough ideals and the upper and lower approximation of their homomorphism images.

Abstract 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 Γsemihyperrings is a generalization of semirings, a generalization of semihyperrings and a generalization of Γ-semirings. 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 Γsemiring 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.