Abstract
In this article, we study the concept of relative bi--hyperideals (-bi--hyperideals) in ordered -semihypergroups and present some related examples of this concept. Especially, characterization of -simple ordered -semihypergroups in terms of -bi--hyperideals is given. Furthermore, we define the notion of -(bi-)int--hyperideals in ordered -semihypergroups and investigate its related properties. We show that -int--hyperideals and --hyperideals coincide in a -regular ordered -semihypergroup.
1. Introduction
One of the motivations for the study of hyperstructures comes from biological inheritance and physical phenomenon as the nuclear fission. Another motivation for the study of hyperstructures comes from chemical reactions and redox reactions. Dehghan Nezhad et al. [1] provided a physical example of hyperstructures associated with the elementary particle physics: leptons. As we know, the Higgs boson is an elementary particle in the standard model of particle physics. In [2], it is shown that the leptons and gauge bosons along with the interactions between their members construct an -structure. Yaqoob et al. [3] studied some properties of (fuzzy) -hyperideals in involution -semihypergroups. In [4], the concepts of uni-soft -hyperideals and uni-soft interior -hyperideals of ordered -semihypergroups are investigated.
Algebraic hyperstructures are a suitable generalization of classical algebraic structures. In a classical algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. The concept of hyperstructures was first introduced by Marty [5] at the eighth congress of Scandinavian Mathematicians in 1934. Nowadays, we can easily find well-written books for the introduction to hyperstructures, which include Corsini [6], Corsini and Leoreanu [7], Davvaz [8], Davvaz and Leoreanu-Fotea [9], Davvaz and Vougiouklis [10], and Vougiouklis [11]. For the information about hyper rings, we refer the reader to Ref. 9.
The study of ordered semihypergroups began with the work of Heidari and Davvaz [12]. In 2015, Davvaz et al. [13] discussed the notion of a pseudo-order in an ordered semihypergroup. The focus of the study was to find out if there is a relationship between ordered semihypergroups and ordered semigroups by using pseudo-orders. In 2016, Gu and Tang [14] answered to the open problem given by Davvaz et al. [13]. In [15], Tipachot and Pibaljommee introduced the concept of fuzzy interior hyperideals on ordered semihypergroups. Recently, Mahboob et al. [16] studied the concept of -hyperideals on ordered semihypergroups. Recall, from Ref. 12, that an ordered semihypergroup is a semihypergroup together with a (partial) order relation that is compatible with the hyperoperation , meaning that, for any ,
Here, means that for any , there exists such that , for all nonempty subsets A and B of S.
Good and Hughes [17] introduced the notion of bi-ideals of a semigroup as early as 1952. In 1962, Wallace [18] introduced the notion of relative ideals (-ideals) on semigroups. In 1967, Hrmová [19] generalized the notion of -ideal by introducing the notion of a -ideal of a semigroup S. Recently, Khan and Ali [20] introduced the concept of relative bi-ideals in ordered semigroups. The notion of -semigroup was introduced by Sen and Saha [21] in 1986, which is a generalization of semigroups. In 2011, Anvariyeh et al. [22] introduced the notion of -hyperideal of a -semihypergroup. Later on, Yaqoob and Aslam [23] studied prime bi--hyperideals of -semihypergroups. Omidi et al. [24, 25] discussed some important properties of bi--hyperideals of an ordered -semihypergroup. Bi--hyperideal is a special case of --hyperideal [26]. In 2017, Tang et al. [27] considered and proved some theorems on fuzzy interior -hyperideals in ordered -semihypergroups. Pseudo-orders are the bridge between ordered -semigroups and ordered -semihypergroups, see Ref. 28.
After an introduction, in Section 2, we introduce some notation and terminologies. Section 2 aims for summarizing the fundamental materials on ordered -semihypergroups. Section 3 is devoted to the study of relative -hyperideals (--hyperideals) and relative bi--hyperideals (-bi--hyperideals) of an ordered -semihypergroup. In this section, our main results are stated and proved. -simple ordered -semihypergroups are characterized by using the notions of --hyperideals and -bi--hyperideals. Finally, in Section 4, the notion of -(bi-)int--hyperideals (-(bi-)interior--hyperideals) are studied and their related properties are discussed. It is shown that, in -regular ordered -semihypergroups the --hyperideals and the -int--hyperideals coincide.
2. Preliminaries
Let S and be two nonempty sets. Then, S is called a -semihypergroup [22] if every is a hyperoperation on S, i.e., , for every , , and , we have
If every is an operation, then S is a -semigroup. Let A and B be two nonempty subsets of S. We define
Also,
In the following, we recall the notion of an ordered -semihypergroup, and then we present basic definitions and notations, which we will need in this article. Throughout this article, unless otherwise specified, S is always an ordered -semihypergroup .
Definition 2.1 (see [29]). An algebraic hyperstructure is called an ordered -semihypergroup if is a -semihypergroup and is a partially ordered set such that for any and , implies and . Here, if A and B are two nonempty subsets of S, then we say that if, for every , there exists such that .
Let S be an ordered -semihypergroup. By a sub -semihypergroup of S, we mean a nonempty subset A of S such that for all and . A nonempty subset A of S is called idempotent if .
Example 1. (See Ref. 25.) Let be an ordered semihypergroup and a nonempty set. We define for every and . Then, is an ordered -semihypergroup.
Let
be a nonempty subset of an ordered
-semihypergroup
. If
His a nonempty subset of
, then we define
Note that if
, then we define
If
Aand
Bare nonempty subsets of
S, then we have
for all
If , then
An element a of an ordered -semihypergroup is regular [25] if there exist and such that . This is equivalent to saying that , for each . An ordered -semihypergroup S is said to be regular if every element of S is a regular element.
Definition 2.2. Let be an ordered -semihypergroup. A nonempty subset A of S is called a left (resp. right) -hyperideal [24] of S if it satisfies the following conditions:
(resp. )
If , , and , then
If A is both a left -hyperideal and a right -hyperideal of S, then it is called a -hyperideal (or two-sided -hyperideal) of S.
3. Basic Properties of Relative Bi--Hyperideals (-Bi--Hyperideals)
Let
be an ordered
-semihypergroup and
and
be the nonempty subsets of
S. Then,
is called a left
-
-hyperideal of
Sif it satisfies the following conditions:
When and such that , it implies that
A right --hyperideal of an ordered -semihypergroup S is defined in a similar way. By two-sided --hyperideal or simply --hyperideal, we mean a nonempty subset of S which is both left and right --hyperideal of S.
Let
and
be the nonempty subsets of
S. A nonempty subset
of
Sis said to be an
-
-hyperideal of
Sif it satisfies the following conditions:
and
When and such that , it implies that
Example 2. Let and be the sets of binary hyperoperations defined as follows:Then, S is a -semihypergroup [23]. We have as an ordered -semihypergroup, where the order relation is defined byThe covering relation and the figure of S are given byLet , and . One can check that is a --hyperideal of S.Here are some elementary properties of these concepts.
Lemma 3.2. Let be an ordered -semihypergroup. If is a sub -semihypergroup of S, then is a --hyperideal of S for each .
Proof. Let . We show that is a --hyperideal of S. We haveSimilarly, we have . Now, let and such that . Then, for some . Hence, and so . Therefore, is a --hyperideal of S.
Theorem 3.3. Let I be a left -hyperideal of an ordered -semihypergroup and such that . If M is an idempotent left --hyperideal of I, then M is a left -hyperideal of S.
Proof. Clearly, I is an ordered sub--semihypergroup of S. We haveIf and such that , then we have . Since M is a left --hyperideal of I, it follows that . Hence, M is a left -hyperideal of S. ∎
Theorem 3.4. Let I be a -hyperideal of a regular ordered -semihypergroup and , such that . Then, any --hyperideal of I is a -hyperideal of S.
Proof. Let I be a -hyperideal of S. Then, . So, I is an ordered sub--semihypergroup of S. Let A be a --hyperideal of I. We prove that A is a -hyperideal of S. Let . Then, there exist and such thatSince I is a -hyperideal of S, it follows thatHence, , for some . This means that . Therefore, I is a regular ordered sub--semihypergroup of S.
Let and . Then, where . Now, suppose that . Then, there exist and such thatSimilarly, we have .
If and such that , then we have . Since A is a --hyperideal of I, it follows that . Hence, A is a -hyperideal of S.
Let a be an element of an ordered -semihypergroup . We denote by (resp. , ) the left (resp. right, two-sided) relative -hyperideal of S generated by a. The intersection of all --hyperideals of S containing the element a is denoted by , where .
Let a be an element of an ordered
-semihypergroup
and
and
be two sub-
-semihypergroups of S. Then,
, where
Proof. Since and , it follows that . We haveOn the contrary, we have . Thus, is a left --hyperideal of S containing a. This means that .
Now, we show that is the smallest left --hyperideal of S containing a. Suppose that A is a left --hyperideal of S containing a. We haveThis proves that (1) holds. Conditions (2) and (3) are proved similarly.
Let be an ordered -semihypergroup and and be nonempty subsets of S. Then, S is said to be left -simple if it has no proper left --hyperideal. In the same way, we can define a right -simple ordered -semihypergroup. If S is a left -simple and right -simple) ordered -semihypergroup, then S is a , -simple ordered -semihypergroup.
Let
be an ordered
-semihypergroup and
and
be sub-
-semihypergroups of
Ssuch that
. Then, the following assertions hold:
S is left -simple if and only if for each
S is right -simple if and only if for each
S is (, )-simple if and only if for each
Proof. The proof is straightforward.
We continue this section with the following definition.
Let
be an ordered
-semihypergroup and
a nonempty subset of
S. A sub-
-semihypergroup
of
Sis called a relative bi-
-hyperideal (
-bi-
-hyperideal) of
Sif the following conditions hold:
When and such that , it implies that
Example 3. We come back to Example 2 and consider ordered -semihypergroup . Let and . It is a routine matter to verify that is a -bi--hyperideal of S.
Lemma 3.8. The intersection of any family of -bi--hyperideals of an ordered -semihypergroup is a -bi--hyperideal of S.
Proof. This proof is straightforward.
Let be an ordered -semihypergroup and be any nonempty subset of S. Then, S is said to be -regular if, for each , there exist and such that .
Let
be an ordered
-semihypergroup and
a sub-
-semihypergroup of
S. Then, the following assertions are equivalent:
S is -regular
for every -bi--hyperideal of S
Proof. Assume that (1) holds. Since is a -bi--hyperideal of S, we get . Thus, . Now, let . Since S is -regular, there exist and such thatHence, and so . Therefore, . Let be a right --hyperideal and a left --hyperideal of S. By routine checking, we can easily verify that and are -bi--hyperideals of S. By Lemma 3.8, is a -bi--hyperideal of S. By hypothesis, we haveLet . Since and , it follows that . By (1) and (2) of Lemma 3.5, we haveThen, , for some . If , then . So, . Therefore, S is -regular. If , then , for some and . Thus, . Therefore, S is -regular.
Let
be an ordered
-semihypergroup and
and
be the nonempty subsets of
S. A sub-
-semihypergroup
of
Sis said to be a
-bi-
-hyperideal of
Sif it satisfies the following conditions:
When and such that , it implies that
Example 4. Let and be the sets of binary hyperoperations defined as follows:Then, S is a -semihypergroup [23]. We have as an ordered -semihypergroup, where the order relation is defined byThe covering relation and the figure of S are given byLet , , and . It is easy to see that is a -bi--hyperideal of S.
The concept of -bi--hyperideals of an ordered -semihypergroup is a generalization of the concept of --hyperideals (left --hyperideals and right --hyperideals) of an ordered -semihypergroup. Obviously, every left (right) --hyperideal of an ordered -semihypergroup S is a -bi--hyperideal of S, but the following example shows that the converse is not true in the general case.
Example 5. Consider the ordered -semihypergroup given in Example 2. It is easy to check that is a -bi--hyperideal on S, where , but it is not a right --hyperideal on S. Since and , but which implies that does not hold.
Theorem 3.11. Let be an ordered -semihypergroup and and be two sub--semihypergroups of S. Then, S is left -simple and right -simple if and only if S does not contain proper -bi--hyperideals.
Proof. Let S be a left -simple and right -simple ordered -semihypergroup and B a -bi--hyperideal of S. It is sufficient to prove that . Consider and . Since S is left -simple, we obtainby Lemma 3.5. We need to consider only two cases:
Case 1. Let . As B is -bi--hyperideal, we have .
Case 2. Let , for some and . By assumption, S is a right -simple ordered -semihypergroup. Therefore,by Lemma 3.5. Since , we have or for some and . By Lemma 3.6, we haveand sowhere . Hence, S is a -regular ordered -semihypergroup. So, there exist and such that . We now turn to the case . From this, we conclude that . So, we obtainSince , it follows that . This gives . If , thenand so . We thus get .
Conversely, suppose that S does not contain proper -bi--hyperideals. Let M be a left --hyperideal and right --hyperideal of S. We haveHence, M is a -bi--hyperideal of S. By assumption, we have . Therefore, S is a left -simple and right -simple ordered -semihypergroup.
4. Relative (Bi)-Int--Hyperideals (-(Bi)-Int--Hyperideals)
Let
be an ordered
-semihypergroup and
a nonempty subset of
S. A sub-
-semihypergroup
of
Sis called a
-int-
-hyperideal of
Sif the following conditions hold:
If , , and , then
It is not difficult to see that every --hyperideal of an ordered -semihypergroup S is a -int--hyperideal of S. The following example shows that the converse is not true in general.
Example 6. Let and . We defineThen, S is a -semihypergroup. We have is an ordered -semihypergroup, where the order relation is defined byThe covering relation and the figure of S are given byLet and . Here, is a -int--hyperideal of S, but not a --hyperideal of S. Indeed, since , it follows that is not a --hyperideal of S.The following example shows that a -regular ordered -semihypergroup is not regular in general.
Example 7. Consider the -semihypergroup given in Example 6. Let be the relation on S defined as follows:Then, is an ordered -semihypergroup. The covering relation and the figure of S are given by:Let . An easy computation shows that S is a -regular ordered -semihypergroup, but it is clearly not regular.
Theorem 4.2. Let be a -regular ordered -semihypergroup, where is a nonempty subset of S. Then, every -int--hyperideal of S is a --hyperideal of S.
Proof. Let be a -int--hyperideal of S. Then, is a sub--semihypergroup of S and . Let . Since S is -regular, there exist and such that . Now, let and . Then,Thus, . By a similar argument, we have . Hence, the result follows.In the following, we introduce the notion of -bi-int--hyperideals as a generalization of --hyperideals, -bi--hyperideals, and -int--hyperideals of ordered -semihypergroups.
Let
be an ordered
-semihypergroup and
a nonempty subset of
S. A sub-
-semihypergroup
of
Sis called a
-bi-int-
-hyperideal of
Sif the following conditions hold:
If , , and , then
Let
be an ordered
-semihypergroup and
a nonempty subset of
S. Then, the following statements hold:
Every --hyperideal of S is a -bi-int--hyperideal of S
The intersection of -bi-int--hyperideals of S is a -bi-int--hyperideal of S
If B is a -bi--hyperideal and C a -int--hyperideal of S, then is a -bi-int--hyperideal of S
Proof. (1) Let A be a --hyperideal of S. Then, and . We haveandTherefore, A is a -bi-int--hyperideal of S.
(2) The proof is similar to the proof of Lemma 3.8.
(3) Clearly, is a sub--semihypergroup of S. We haveand
Hence, . Therefore, is a -bi-int--hyperideal of S.
Example 8. Let and . We defineThen, S is a -semihypergroup. We have as an ordered -semihypergroup [29], where the order relation is defined byThe covering relation and the figure of S are given byLet and . Then, and , so . Hence, is a -bi-int--hyperideal of S. It is easy to see that () is not a --hyperideal of S.
5. Conclusion
In this article, we studied some properties of -(bi-)-hyperideals of ordered -semihypergroups. In particular, we introduced and studied -int--hyperideals and -bi-int--hyperideals. Furthermore, we proved that -int--hyperideals and --hyperideals coincide in -regular ordered -semihypergroups. When we deal with -(bi-)-hyperideals of ordered -semihypergroups, it is natural to talk about fuzzy -(bi-)-hyperideal. According to the research results, it is suggested to define and investigate some properties of fuzzy -(bi-)-hyperideals, rough prime -bi--hyperideals, fuzzy prime -bi--hyperideals, and uni-soft -int--hyperideals in ordered -semihypergroups. As an application of the results of this article, the corresponding results of ordered semihypergroups can be also obtained by moderate modification.
Funding
This work was supported by the National Key R&D Program of China (Grant no. 2019YFA0706402), the Natural Science Foundation of Guangdong Province (Grant no. 2018A0303130115), and Guangzhou Academician and Expert Workstation (Grant no. 20200115-9).
Statements
Data availability statement
All datasets presented in this study are included in the article/supplementary material.
Author contributions
All authors have made a substantial, direct, and intellectual contribution to the work and approved it for publication.
Conflict of interest
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
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Summary
Keywords
ordered -semihypergroup, -bi- -hyperideal, -(bi-)int- -hyperideal, -regular AMS mathematics subject classification: 16Y99, 20N20, 06F99
Citation
Rao Y, Xu P, Shao Z, Kosari S and Omidi S (2020) Some Properties of Relative Bi-(Int-)Γ-Hyperideals in Ordered Γ-Semihypergroups. Front. Phys. 8:555573. doi: 10.3389/fphy.2020.555573
Received
25 April 2020
Accepted
24 August 2020
Published
05 November 2020
Volume
8 - 2020
Edited by
Muhammad Javaid, University of Management and Technology, Lahore, Pakistan
Reviewed by
Kazuharu Bamba, Fukushima University, Japan
Naveed Yaqoob, Majmaah University, Saudi Arabia
Updates
Copyright
© 2020 Rao, Xu, Shao, Kosari and Omidi.
This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: S. Kosari, saeedkosari38@yahoo.com
This article was submitted to Mathematical and Statistical Physics, a section of the journal Frontiers in Physics
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