Abstract

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

1. 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 -semihypergroups, we may refer to Ref. [3].

The 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. The notion of hypergroups was initially founded by F. Marty [7] in 1934.

Recently, many authors, for example, those in Refs. [8, 9], have investigated on ordered hyperstructures. The paper given in Ref. [8] is a detailed study of interior hyperfilters in ordered -semihypergroups. In Ref. [9], w-pseudo-orders in ordered (semi)-hyperrings were defined, and some important properties are investigated.

The notion of uni-soft interior -hyperideals is investigated in Ref. [10]. Motivated by these studies, this note investigates the ordered -semihypergroups that their interior hyperideal is equal to the hyperideal. We prove that in regular (resp., intraregular, semisimple) ordered -semihypergroups, the concepts of interior -hyperideals and -hyperideals coincide.

Definition 1 (see [11]). Let and be two nonempty sets. Then, is called a -semihypergroup if every is a hyperoperation on , i.e., for every , and for every and , we have .
Let and be two nonempty subsets of . We define

Definition 2. An ordered -semihypergroup is a -semihypergroup together with a partial order relation such that for any and , we haveHere, means that for any , there exists such that , where .
Now, let
.
Then, can be called as follows:(1)Regular (resp., intraregular) if (resp., ) for every (2) is called semisimple if for every A nonempty subset of is called a -hyperideal of if(1) and (2)

Definition 3 (see [5]). A sub -semihypergroup of is called an interior -hyperideal (briefly, --hyperideal) if(1)(2)

Remark 1. Note that each hyperideal of an ordered hyperstructure is an --hyperideal, but an --hyperideal need not be hyperideal.

Example 1. Let and . Define the hyperoperation (as shown in Table 1) and (partial) order relation on as follows:Here, is an --hyperideal of ordered -semihypergroup but not a -hyperideal of . Indeed, as and , is not a -hyperideal of .
In this note, we investigate on the ordered -semihypergroups that their interior hyperideal is equal to the hyperideal.

Table 1 
Table of for Example 1.

2. Main Results

This section aims to outline sufficient conditions for an --hyperideal to be a -hyperideal. We continue our study with the characterization of regular (resp., Intraregular, semisimple) ordered -semihypergroup in terms of --hyperideals.

Theorem 1. Let be regular. Then, every --hyperideal of is a -hyperideal.

Proof. Assume that is an --hyperideal of and . By hypothesis, there exist and such that . It means that . If and , thenThus, . Similarly, .

Example 2. Consider the -semihypergroup [12] (see Tables 2 and 3).
Now, we setClearly, is regular. The only --hyperideals of are and . Both the --hyperideals are -hyperideal.

Table 2 
Table of for Example 2.
Table 3 
Table of for Example 2.

Theorem 2. Let be intraregular. Then, we get those as follows:(1)Every --hyperideal of is a -hyperideal(2)Every --hyperideal of is idempotent

Proof. (1)Assume that is an --hyperideal of and . By hypothesis, there exist and such that . It means that . If and , then So, . Similarly, .(2)Assume that is an --hyperideal of . Then, we haveNow, let . Then, for some and . By hypothesis, there exist and such that . We haveThus, and so .

Example 3. Consider the -semihypergroup [13] (see Tables 4 and 5).
Now, we setClearly, is an intraregular ordered -semihypergroup. The only --hyperideals of are and . Both the --hyperideals are -hyperideal and idempotent.

Table 4 
Table of for Example 3.
Table 5 
Table of for Example 3.

Theorem 3. Let be a semisimple ordered -semihypergroup. Then, every --hyperideal of is a -hyperideal.

Proof. Assume that is an --hyperideal of and . By hypothesis, there exist and such that . It means that . If and , thenSo, . Similarly, .

Theorem 4. is semisimple if and only if every -hyperideal of is idempotent.

Proof. (Necessity). Let be a -hyperideal of . By hypothesis, we haveAlso,So, , and it completes the proof. Sufficiency. Let . We denote by the -hyperideal of generated by . Then, we get .
By hypothesis, we haveTherefore, is semisimple.

Example 4. In Example 2,is a partial order relation. Clearly, is semisimple. The only --hyperideals of are and . Both the --hyperideals are -hyperideal and idempotent.

3. Conclusions

This paper gives some conditions under which the --hyperideals are -hyperideals. By Theorems 1–3, we prove that in a regular (resp., intraregular, semisimple) ordered hyperstructure , every interior hyperideal of is a hyperideal. By Theorems 3 and 4, is a semisimple ordered hyperstructure if and only if every interior hyperideal of 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

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was funded by the National Natural Science Foundation of China (No. 62172116, 61972109), and the Guangzhou Academician and Expert Workstation (No. 20200115-9).