Solvability of a triple of binary relations with an application to green’s relations

In this paper, we introduce the notion of a solvable triple of binary relations on a set. This notion generalizes the notion of a regular relation and all other notions that are variants of the notion of the regularity, defined previously by many people. We also give some characterizations of the solvability of a triple of relations and use this to study Green’s relations on the monoid of binary relations on a set.

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.

On the g-hypergroupoids associated with g-hypergraphs

In this paper, we associate a partial g-hypergroupoid with a given g-hypergraph and analyze the properties of this hyperstructure. We prove that a g-hypergroupoid may be a commutative hypergroup without being a join space. Next, we define diagonal direct product of g-hypergroupoids. Further, we construct a sequence of g-hypergroupoids and investigate some relationships between it?s terms. Also, we study the quotient of a g-hypergroupoid by defining a regular relation. Finally, we describe fundamental relation of an Hv-semigroup as a g-hypergroupoid.

A Preliminary Approach to the Relationship between the Rockburst Proneness and the Spectrum Characteristic of Rock

How to judge the rockburst proneness of rocks is one of the key works to predict rockburst. A preliminary approach is made to the relationship between the rockburst proneness and the spectrum characteristic of the rocks by means of experiments in the present paper. The following conclusions are obtained. Firstly, under the condition without loading, there is not any regular relation between the rockburst proneness and the spectrum characteristic of the tested rocks. And secondly, under the condition of uniaxial loading, the spectrum curves of the tested rocks with different rockburst proneness all change with the increase of stress. These curves possess similar tendencies, but different shapes, inflexions and developing rates.