The concept of almost interior Γ -hyperideals ( A − I − Γ -hyperideals) in ordered Γ -semihypergroups is a generalization of the concept of interior Γ -hyperideals ( I − Γ -hyperideals). In this study, the connections between I − Γ -hyperideals and A − I − Γ -hyperideals in ordered Γ -semihypergroups were presented. Also, we define the notion of m , n - Γ -hyperfilters of ordered Γ -semihypergroups and provide useful results on it. Moreover, we use the weak pseudoorders to construct quotient ordered Γ -semihypergroups by using ordered regular equivalence relations. These concepts lead us to a new research direction in ordered Γ -semihypergroups.

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.

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.

In this study, we propose the concept of factorizable ordered hypergroupoids (semihypergroups) and present several of its properties. Our goal is to construct ordered hypergroupoid from blood groups together with some other information. Finally, we discuss right magnifying elements for further research.

Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by human, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real-life, problems being often ambiguous, so it is very challenging for an expert to exemplify those problems through applying an FG. Vague graph structure (VGS), belonging to FGs family, has good capabilities when facing with problems that cannot be expressed by FGs. VGSs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. Therefore, in this paper, we apply the concept of vague sets (VSs) to GS. We define certain notions, VGS, strong vague graph structure (SVGS), and vague β i -cycle and describe these notions by several examples. Likewise, we introduce ψ -complement, self-complement (SC), strong self-complement (SSC), and totally strong self-complement (TSSC) in VGS and investigate some of their properties. Finally, an application of VGS is presented.

In the present paper, we will concentrate our efforts on ordered Krasner hyperrings and investigate some of their related properties. Moreover, we introduce and analyze the notion of interior hyperideal in ordered Krasner hyperrings. We also characterize intra-regular ordered Krasner hyperrings by the properties of these interior hyperideals. Finally, we give some results on ordered Krasner hyperrings.