In the present paper, we introduce the notion of k-hyperideals on ordered semihyperrings. Then, we investigate some fundamental properties of k-hyperideals of ordered semihyperrings. Indeed, we define (m, n)-hyperideals and (m, n)-bi-hyperideals in ordered semihyperrings and investigate some of their related properties. Moreover, we introduce and analyze the notion of prime (m, n)-bi-hyperideal of an ordered semihyperring.
In this article, we study the concept of relative bi-Γ-hyperideals (J-bi-Γ-hyperideals) in ordered Γ-semihypergroups and present some related examples of this concept. Especially, characterization of J-simple ordered Γ-semihypergroups in terms of J-bi-Γ-hyperideals is given. Furthermore, we define the notion of J-(bi-)int-Γ-hyperideals in ordered Γ-semihypergroups and investigate its related properties. We show that J-int-Γ-hyperideals and J-Γ-hyperideals coincide in a J-regular ordered Γ-semihypergroup.
In this paper, we reformulate several results in commutative algebra in terms of commutative hyperrings. We introduce n-hyperideals in commutative hyperrings and give its some basic properties. Based on new definitions and theorems, we obtain some results in the hyperring theory. Also, the paper is stated a characterization for fundamental n-hyperideals.
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.
The concept of $\Gamma$-semihypergroups is a generalization of semigroups, a generalization of semihypergroups and a generalization of $\Gamma$-semigroups. In this paper, we study the notion of bi-$\Gamma$-hyperideals in ordered $\Gamma$-semihypergroups and investigate some properties of these bi-$\Gamma$-hyperideals. Also, we define and use the notion of regular ordered $\Gamma$-semihypergroups to examine some classical results and properties in ordered $\Gamma$-semihypergroups.
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