Finitely generated rings obtained from hyperrings through the fundamental relations

In this article, we introduce and analyze a strongly regular relation $\omega^{*}_{\mathcal{A}}$ on a hyperring$R$ such that in a particular case we have $|R/\omega^{*}_{\mathcal{A}}|\leq 2$ or$R/\omega^{*}_{\mathcal{A}}=<\omega^{*}_{\mathcal{A}}(a)>$, i.e., $R/\omega^{*}_{\mathcal{A}}$ is a finite generated ring. Then, by using the notion of $\omega^{*}_{\mathcal{A}}$-parts, we investigate the transitivity condition of $\omega_{\mathcal{A}}$. Finally, we investigate a strongly regular relation $\chi^{*}_{\mathcal{A}}$ on the hyperring $R$ such that $R/\chi^{*}_{\mathcal{A}}$ is a commutative ring with finite generated.

The commutative quotient structure of m-idempotent hyperrings

The α* -relation is a fundamental relation on hyperrings, being the smallest strongly regular relation on hyperrings such that the quotient structure R/α* is a commutative ring. In this paper we introduce on hyperrings the relation ζm, which is smaller than α*, and show that, on a particular class of m-idempotent hyperrings R, it is the smallest strongly regular relation such that the quotient ring R/ζ*m is commutative. Some properties of this new relation and its differences from the α* -relation are illustrated and discussed. Finally, we show that ζm is a new representation for α* on this particular class of m-idempotent hyperrings.

Fundamental relation on m-idempotent hyperrings

The γ*-relation defined on a general hyperring R is the smallest strongly regular relation such that the quotient R/γ* is a ring. In this note we consider a particular class of hyperrings, where we define a new equivalence, called $\varepsilon^{*}_{m} $, smaller than γ* and we prove it is the smallest strongly regular relation on such hyperrings such that the quotient R/ $\varepsilon^{*}_{m} $ is a ring. Moreover, we introduce the concept of m-idempotent hyperrings, show that they are a characterization for Krasner hyperfields, and that $\varepsilon^{*}_{m} $ is a new exhibition for γ* on the above mentioned subclass of m-idempotent hyperrings.

Convex ordered Gamma-semihypergroups associated to strongly regular relations

In this study, we introduce and investigate the notion of convex ordered Gamma-semihypergroups associated to strongly regular relations. Afterwards, we prove that if sigma is a strongly regular relation on a convex ordered Gamma-semihypergroup, then the quotient set is an ordered Gamma-sigma-semigroup. Also, some results on the product of convex ordered Gamma-semihypergroups are given. As an application of the results of this paper, the corresponding results of ordered semihypergroups are also obtained by moderate modifications.