In this article, we introduce the notion of quasi-hyperideal in semihypergroups, and moreover, we introduce the notions of an (m, n)-quasi-hyper-ideal, n-right hyperideal, and m-left hyperideal in semihypergroups, and relations between them are studied. We obtain different characterizations concerning different properties of (m, n)-quasi-hyperideals, minimal (m, n)-quasi-hyperideals, minimal m-left hyperideals, and minimal n-right hyperideals, and relations between them are investigated. Also, some intersection properties and characterizations of (m, n)-quasi-hyperideals of semihyperideals and regular semihyperideals have been studied. In the sequel, m-left simple, n-right simple, and (m, n)-quasi-simple semihypergroups are defined, and some properties of them are investigated.

This paper deals with a class of algebraic hyperstructures called left almost semihypergroups (LA-semihypergroups), which are a generalization of LA-semigroups and semihypergroups. We introduce the notion of LA-semihypergroup, the related notions of hyperideal, bi-hyperideal, and some properties of them are investigated. It is a useful nonassociative algebraic hyperstructure, midway between a hypergroupoid and a commutative hypersemigroup, with wide applications in the theory of flocks, and so forth. We define the topological space and study the topological structure of LA-semihypergroups using hyperideal theory. The topological spaces formation guarantee for the preservation of finite intersection and arbitrary union between the set of hyperideals and the open subsets of resultant topologies.

In this article, we study Γ-semihypergroups introduced recently. We introduce the hyper versions of Green's relations in Γ-semihypergroups and give some related characterizations. We introduce the n-prime Γ-hyperideal and n-semiprime Γ-hyperideal of a Γ-semihypergroup and show that for any integer n ≥ 2, n-prime Γ-hyperideals are a generalization of prime Γ-ideals. We also give the relationship between n-prime Γ-hyperideals and Γ-hyperideal extensions in Γ-semihypergroups. Then, we deal with prime ideal and introduce the notion of prime radical in a Γ-semihypergroup M, and also, we obtain some results and relations among the prime radicals of M and S, where S is the left operator semihypergroup of M.

Krasner Hyperrings are an important class of algebraic hyperstructures which generalize rings further to allow multiple output values for the addition operation. Krasner (m, n)-hyperrings are a generalization of canonical n-ary hypergroups, a generalization of (m, n)-rings, a generalizations of hyperrings in the sense of Krasner and a subclass of the (m, n)-hyperrings. Badawi introduced a generalization of prime ideals called 2-absorbing ideals and this idea is further generalized in a paper by Anderson and Badawi to a concept called n-absorbing ideals. In this paper, we generalize this notion in Krasner (m, n)-hyperrings introducing and studying the notion of (k, n)-absorbing hyperideal. Several properties of them are provided. Also we introduce the notion of (k, n)-absorbing primary hyperideals of H generalizing the analogs results obtained in rings. Several properties of them are provided.

We give some characterizations of quasi-ideals in Γ-semigroups and some characterizations of the regularity of a Γ-semigroup in terms of left, right, quasi-and bi-ideals. Also, we introduce and give some characterizations of semiprime and quasi-reflexive Γ-semigroups in terms of minimal and canonical quasi-ideals generalizing those for semigroups.