Characterizations of Hyperideals and Interior Hyperideals in Ordered Γ -Semihypergroups

We give some conditions on ordered Γ -semihypergroups under which their interior hyperideal is equal to the hyperideal. In this paper, it is shown that in regular (resp., intraregular, semisimple) ordered Γ -semihypergroups, the hyperideals and the interior hyperideals coincide. To show the importance of these results, some examples and conclusions are provided.


Introduction and Preliminaries
Heidari and Davvaz [1] gave the idea of an ordered semihypergroup in 2011. Connection between ordered semihypergroups was studied by Tang et al. [2]. For some works on ordered Γ-semihypergroups, we may refer to Ref. [3].
Te general structure of factorizable ordered hypergroupoids is studied in Ref. [4]. Tang et al. [5] and Tipachot and Pibaljommee [6] combined the fuzzy set with ordered hyperstructures and proposed the concept of fuzzy interior hyperideal and proved some results. Te notion of hypergroups was initially founded by F. Marty [7] in 1934.
Te notion of uni-soft interior Γ-hyperideals is investigated in Ref. [10]. Motivated by these studies, this note investigates the ordered Γ-semihypergroups that their interior hyperideal is equal to the hyperideal. We prove that in regular (resp., intraregular, semisimple) ordered Γ-semihypergroups, the concepts of interior Γ-hyperideals and Γ -hyperideals coincide.
Let A and B be two nonempty subsets of H. We defne together with a partial order relation ≤ such that for any h, h ′ , x ∈ H and α ∈ Γ, we have Here, C ⪯ D means that for any c ∈ C, there exists d ∈ D such that c ≤ d, where ∅ ≠ C, D⊆H. Now, let Ten, (H, Γ, ≤ ) can be called as follows: (1) Regular (resp., intraregular) if K⊆(KΓHΓK] (resp., K⊆(HΓKΓKΓ H ( ])) for every K⊆H Note that each hyperideal of an ordered hyperstructure H is an I-Γ-hyperideal, but an I-Γ-hyperideal need not be hyperideal.
Defne the hyperoperation c (as shown in Table 1) and (partial) order relation ≤ on H as follows: In this note, we investigate on the ordered Γ-semihypergroups that their interior hyperideal is equal to the hyperideal.

Main Results
Tis section aims to outline sufcient conditions for an I-Γ -hyperideal to be a Γ-hyperideal. We continue our study with the characterization of regular (resp., Intraregular, semisimple) ordered Γ-semihypergroup in terms of I-Γ -hyperideals.   Tables 2 and 3). Now, we set Now, let a ∈ KΓK ]. Ten, a ⪯ kck ′ for some k, k ′ ∈ K and c ∈ Γ. By hypothesis, there exist h, h ′ ∈ H and μ, λ, δ ∈ Γ such that a ⪯ hμaλaδh ′ . We have

Conclusions
Tis paper gives some conditions under which the I-Γ -hyperideals are Γ-hyperideals. By Teorems 1-3, we prove that in a regular (resp., intraregular, semisimple) ordered hyperstructure H, every interior hyperideal of H is a hyperideal. By Teorems 3 and 4, H is a semisimple ordered hyperstructure if and only if every interior hyperideal of H is idempotent. Our future work will concentrate on some results which are related with the fuzzy interior hyperideals of ordered hyperstructures.

Data Availability
No data were used to support this study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.