In this paper, we introduce and study the concepts of prime left, semiprime left and irreducible left hyperideals in ternary semihypergroups and investigate some basic properties of them. We introduce the concepts of hyperfilter and hypersemilattice congruence of ternary semi- hypergroups. We give some characterizations of hyperfilters in ternary semihypergroups. Some relationships between hyperfilters, prime hyper- ideals and hypersemilattice congruences in ternary semihypergroups are considered. We also introduce the notion of hyperideals extensions in ternary semihypergroups and some properties of them are investigated.

The concept of a Γ-semihyperring is a generalization of a semiring, semihyperring, Γ-semiring. In this Paper we introduce the notion of bi-interior Γ-hyperideals, quasi-interior Γ-hyperideals and bi-quasi-interior Γ-hyperideals in a Γ-semihyperring as a generalization of Γ-hyperideal, left-Γ-hyperideal, right-Γ-hyperideals, bi Γ-hyperideal, quasi Γ-hyperideal, interior Γ-hyperideals of Γ-semihyperring. We studied the properties of these Γ-hyperideals and characterized them in simple Γ-semihyperring and regular Γ-semihyperring

Let R be a multiplicative hyperring. In this paper, we introduce and study the concept of n-absorbing hyperideal which is a generalization of prime hyperideal. A proper hyperideal I of R is called an n-absorbing hyperideal of R if whenever α 1o. . . oα n+1⊆ I for α 1, . . ., α n+1∈ R, then there are n of the α , is whose product is in I.

In a ternary semihyperring, addition is a hyperoperation and multiplication is a ternary operation. Indeed, the notion of ternary semihyperrings is a generalization of semirings. Our main purpose of this paper is to introduce the notions of fuzzy hyperideal and fuzzy bi-hyperideal in ternary semihyperrings. We give some characterizations of fuzzy hyperideals and investigate several kinds of them.

The aim of this article is to study ordered semihypergroups in the framework of (M, N)-int-soft bi-hyperideals. In this paper, we introduce the notion of (M, N)-int-soft bi-hyperideals of ordered semihypergroups. Some properties of (M, N)-int-soft bi-hyperideals in ordered semihypergroups are provided. We show that every int-soft bi-hyperideal is an (M, N)-int-soft bi-hyperideals of S over U but the converse is not true which is shown with help of an example. We characterize left (M, N) simple and completely regular ordered semihyper groups by means of (M, N)-int-soft bi-hyperideals.

‎Krasner $F^{(m,n)}$-hyperrings were introduced and  investigated by Farshi and Davvaz. In this paper, our purpose is to  define and characterize  three particular classes of $F$-hyperideals in a Krasner $F^{(m,n)}$-hyperring, namely prime $F$-hyperideals, maximal $F$-hyperideals and  primary $F$-hyperideals, which extend similar concepts of ring context. Furthermore, we  examine the relations between these structures. Then a number of major conclusions are given to explain the general framework of these structures.

For Krasner hyperrings, we study d-prime hyperideals where d is a homo-derivation. Furthermore, we show that every maximal d-hyperideal and d-prime hyperideal is a prime hyperideal of a commutative hyperring. Finally, we prove that if W is a d-prime hyperideal of a hyperring R and d(qn) ∈ W for some q∈R, then d2(q)∈ W.