We introduce a new Regular relation δ on a given group G and show that δ is a congurence relation on G, with respect to module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly Regular relation on the hypergroup H, then the effect of δ on ρ, results in a strongly Regular relation on H such that its quotient is an abelian group.

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 t-semihyperrings is a generalization of semirings, a generalization of semihyperrings and a generalization of tsemirings. In this paper, we introduce an equivalence relation g* on a t-semihyperrings R and we show that it is strongly Regular. Furthermore, R/g*, the set of all equivalence classes of this relation is a t/b*-semiring. The relation g* is called the fundamental relation and the t-semiring R/g * is called the fundamental semiring. Fundamental relations are the main tools in the study of t-semihyperrings. We present some results about fundamental relations and fundamental semirings. Finally, we show that there is a covariant functor between the category of t-semihyperrings and the category of semirings.

Hypergroups that have at least one identity element and where each element has at least one inverse are called Regular hypergroup. In this regards, for a Regular hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly Regular relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly Regular relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

In this paper we consider a strongly Regular relation q on hypermodules so that the quotient is a module (with abelian group) over a fundamental commutative ring. Also, we state necessary and sufficient conditions so that the relation q is transitive, and finally we prove that q is transitive on hypermodules.

In this paper, by considering the notion of -hyperalgebras for an arbitrary signature  , we study the notions of Regular and strongly Regular relations on a -hyperalgebra, A. We show that each Regular relation which contains a strongly Regular relation is a strongly Regular relation. Then we concentrate on the connection between the fundamental relation of A and the set of complete parts of A.

In this paper, the concepts of m-complete parts in a hyperring are introduced and its connection with complete parts of a hyperring are investigated. Moreover, ε _m-complete hyperrings are defined based on ε _m-relation in hyperrings, and a characterization is obtained by m-complete parts for them.