Abstract
The concept of almost interior -hyperideals (-hyperideals) in ordered -semihypergroups is a generalization of the concept of interior -hyperideals (-hyperideals). In this study, the connections between -hyperideals and -hyperideals in ordered -semihypergroups were presented. Also, we define the notion of --hyperfilters of ordered -semihypergroups and provide useful results on it. Moreover, we use the weak pseudoorders to construct quotient ordered -semihypergroups by using ordered regular equivalence relations. These concepts lead us to a new research direction in ordered -semihypergroups.
1. Introduction
The concept of hyperstructure was first introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians. Moreover, the hyperrings were introduced after the hyperfields, by Krasner [2], for solving some problems in valuation of fields. From the early attempts [1, 2] to recent studies, hyperstructure theory has been used in diverse branches of mathematics [3, 4], physics [5, 6], chemistry [7], biology [8, 9], etc. In [5], Davvaz et al. showed that the leptons and gauge bosons along with the interactions between their members construct a weak algebraic hyperstructure (-structure). Some applications of hyperstructure theory in mathematics, cryptography, codes, and other fields can be found in [10]. In [3], Jun studied algebraic and geometric aspects of Krasner hyperrings in detail. In 2018, Omidi and Davvaz [11] introduced and studied the notion of ordered regular equivalence relations in ordered semihyperrings.
Semihypergroups are as important in algebraic hyperstructures as the semigroups in algebraic structures. Recently, Daengsaen et al. [12] studied minimal and maximal hyperideals of n-ary semihypergroups. Ordered semihypergroups were introduced by Heidari and Davvaz [13] as a generalization of the ordered semigroups. In 2015, Davvaz et al. [14] presented a connection between ordered semihypergroups and ordered semigroups by using pseudoorder. In 2016, Gu and Tang [15] studied an open problem for ordered semihypergroups and established a partial solution. Next, Tang et al. [16] studied the further properties of ordered regular equivalence relations in ordered semihypergroups. Moreover, the authors gave a complete answer of the open problem given by Davvaz et al. in [14].
As a generalization of a semihypergroup, Anvariyeh et al. [17], in 2010, introduced the notion of -semihypergroups and discussed the -hyperideals of -semihypergroups. In [18], the concept of (fuzzy) -hyperideals of involution -semihypergroups was investigated. Recently, fuzzy set theory has been well developed in the framework of ordered -semihypergroup theory [19]. In 2020, Rao et al. [20] introduced and studied the concept of relative bi-(int-)-hyperideals in ordered -semihypergroups. The beauty of relative bi-(int-)-hyperideals is that instead of involving all the elements of the ordered -semihypergroup , we deal with the elements belonging to the nonempty subsets of . The concept of -ideals of semigroups was introduced by Grosek and Satko [21] and was further investigated in [22]. In [23], Suebsung et al. studied -hyperideals in semihypergroups.
Hyperfilter theory in ordered hyperstructures has been investigated by many mathematicians. In 2015, Tang et al. [24] studied (fuzzy) hyperfilters in ordered semihypergroups. In 2018, Omidi et al. [25] applied the hyperfilter theory to ordered -semihypergroups and obtained some results in this respect. In 2019, Bouaziz and Yaqoob [26] investigated some properties of rough hyperfilters in po-LA-semihypergroups. The notion of -hyperideals was introduced in various ordered hyperstructures and had been studied by many authors, for instance, Mahboob et al. [27], Omidi and Davvaz [28], and many others.
Motivated by the work of Kaopusek et al. [29] and Mahboob and Khan [30], we applied the concepts of almost interior ideals and -hyperfilters to ordered -semihypergroups. In Section 2, we give the preliminary concepts concerning ordered -semihypergroups and -hyperideals. In Section 3, the notion of almost interior -hyperideals (briefly, -hyperideal) is given. In addition, some properties of -hyperfilters in ordered -semihypergroups have been proven. Finally, we conclude in Section 4 with some arising examples. Theory of -hyperideals and -hyperfilters is useful to explore new results associated with ordered -semihypergroups.
2. Preliminaries
Let be a nonempty set and be the family of all nonempty subsets of . A mapping is called a hyperoperation on . A hypergroupoid is a set together with a (binary) hyperoperation. If and are two nonempty subsets of and , then we denote
A hypergroupoid is called a semihypergroup if for every in ,
The notion of -semihypergroups has been introduced by Anvariyeh et al. [17] in 2010. In this section, we present some notions related to (ordered) -semihypergroup theory.
Definition 1 (see [17]). Let and be two nonempty sets. Then, is said to be 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
Also,
If every is an operation, then is a -semigroup. A nonempty subset of is called a sub -semihypergroup of if for every and .
Definition 2. (see [25]). Let be an ordered relation on a nonempty set . By an ordered -semihypergroup we mean an algebraic hyperstructure in which the following are satisfied:(1) is a -semihypergroup.(2) is a (partially) ordered set.(3)For any and , implies and . If and are nonempty subsets of , thenLet be an equivalence relation on an ordered -semihypergroup . If and are nonempty subsets of , then(1).(2).(3) and and .(4) and .(5).A relation on an ordered -semihypergroup is called a pseudoorder [25] on if (1) ; (2) and imply for all ; (3) implies and for all and .
Theorem 1 (see [25]). Let be an ordered -semihypergroup and a pseudoorder on . Then, there exists a strongly regular equivalence relation on such that is an ordered -semigroup, where .
Let be a nonempty subset of an ordered -semihypergroup . is defined as follows:
For convenience, given , we write . Let and be nonempty subsets of an ordered -semihypergroup . Then,(1).(2)If , then .(3) and .(4).
Let be an ordered -semihypergroup. By a sub -semihypergroup of we mean a nonempty subset of such that . We denote
The interior -hyperideal (in short -hyperideal) is defined as follows.
Definition 3 (see [19]). Let be an ordered -semihypergroup. A sub -semihypergroup of is called an interior -hyperideal (in short -hyperideal) of if(1).(2).
An ordered -semihypergroup is called regular [31] if for every there exist , such that . This is equivalent to saying that , for every or , for every .
3. -Hyperideals and -Hyperfilters
Throughout the rest of this paper: will be an ordered -semihypergroup. We begin this section with the definition of an almost interior -hyperideal (in short -hyperideal) on an ordered -semihypergroup .
Definition 4. Let be an ordered -semihypergroup. A nonempty subset of is called an -hyperideal of if(1) for any .(2).
Example 1. Let and . Define hyperoperations and on by the following tables:Then, is a -semihypergroup [32]. We have that is an ordered -semihypergroup where the (partial) order relation is defined byWe give the covering relation asThe Hasse diagram of is shown in Figure 1.
Here, is an -hyperideal of .
Lemma 1. Let be an ordered -semihypergroup. Then, every -hyperideal of is an -hyperideal of .
Proof. Let be an -hyperideal of . Then, . If , thenIt implies that . So, for any . Therefore, is an -hyperideal of .
In Section 4, we show that the converse of Lemma 1 is not true in general, i.e., an -hyperideal may not be an -hyperideal of .
Lemma 2. Let be an ordered -semihypergroup. If and are -hyperideals of , then is an A-I-Gamma-hyperideal of S.
Proof. Clearly, . If , thenSo, . It implies thatSince is an -hyperideal of , it follows that . This means that . Therefore, is an -hyperideal of .
Theorem 2. Let be an ordered -semihypergroup and . Then, the following assertions are equivalent:(1) has no proper -hyperideal.(2)For any , there exists such that .
Proof. (1): Assume that (1) holds. Let . Then, is not a -hyperideal. So, there exists such that . It implies that .(2): Suppose that for any , there exists such that . This givesTherefore, is not an -hyperideal for every . Now, let be a proper -hyperideal of . Then, for some . Thus, , and hence . Since is an -hyperideal of , it follows that . So, . This means that is an -hyperideal of , which is a contradiction. This completes the proof.
In the following, as a continuation of the previous study [25], the authors discuss -hyperfilter in ordered -semihypergroups. The -hyperfilter of ordered -semihypergroups is defined as follows.
Definition 5. A sub -semihypergroup of an ordered -semihypergroup is called a left -hyperfilter (resp. right -hyperfilter) of if(1)For all and (resp. ).(2)For all and .Here, are positive integers. Note that if is both a left -hyperfilter and a right -hyperfilter of , then is called an -hyperfilter of .
Let be a sub -semihypergroup of an ordered -semihypergroup . We defineFor , we write instead of . Note that condition (2) in Definition 5 is equivalent to .
The concept of an -hyperfilter is a generalization of the concept of a -hyperfilter of . An -hyperfilter of is a -hyperfilter of for all and . For , is a -hyperfilter of . The following example shows that any -hyperfilter of an ordered -semihypergroup need not always be a -hyperfilter of . Also, see Example 4 in Section 4.
Example 2. Let and . We defineThen, is a -semihypergroup. We have that is an ordered -semihypergroup where the order relation is defined byWe give the covering relation asThe figure of is shown in Figure 2.
Let . Clearly, is an -hyperfilter of for . But is not a -hyperfilter of . Indeed: since or , or for any , for and , we have that is an -hyperfilter of for but not a -hyperfilter of .
Lemma 3. Let be a family of -hyperfilters of an ordered -semihypergroup . If , then is an -hyperfilter of .
Proof. Let . Then, for each . Since is a sub -semihypergroup of for each , it follows that for all . Hence, is a sub -semihypergroup of . Now, let , and . Then, there exists for some . Since , it follows that for each . Since is an -hyperfilter of for each , we get for each . It implies that . Now, let and . Then, for each . Since is an -hyperfilter of for all , it follows that for all . So, . Therefore, is an -hyperfilter of .
One of the distinguished properties of the -hyperfilters is that their union is not an -hyperfilter in general, for example, see [25, 30].
Let be an ordered -semihypergroup for all . Define byfor all , and . Now, we put if and only if, for all , .
Then, is an ordered -semihypergroup [33]. In the following, we study the behavior of -hyperfilters on the product of ordered -semihypergroups.
Theorem 3. Let be an -hyperfilter on the ordered -semihypergroup for all . Then, is an -hyperfilter on .
Proof. We divide the proof into three steps. Step 1. We first show that is a sub -semihypergroup of . Let . Then, for each . As ’s is a sub -semihypergroup of , we have . So, we have Thus, is a sub -semihypergroup of . Step 2. Now, let and . Then, Step 3. Let and such that . Then, for all , we have . Since is an -hyperfilter of for each , we get for each . It implies that . Therefore, is an -hyperfilter of .
A mapping of an ordered -semihypergroup into an ordered -semihypergroup is said to be a normal -homomorphism if (1) for all , and ; (2) is isotone, i.e., for any , implies .
Theorem 4. Let be a normal -homomorphism of ordered -semihypergroups and . If is an -hyperfilter of , thenis an -hyperfilter of .
Proof. We divide the proof into three steps. Step 1. Let , , and . Then, . Since is a sub -semihypergroup of and is normal -homomorphism, we have It implies that . Thus, is sub -semihypergroup of . Step 2. Let , , and . Then, Step 3. Now, suppose that and such that . Then, . Since and is normal -homomorphism, we have . Since is an -hyperfilter of , we get . So, . Therefore, is an -hyperfilter of . Let be a sub -semihypergroup (resp. nonempty subset) of an ordered -semihypergroup . Then, is called an -hyperideal (resp. generalized -hyperideal) [34] of if (1) and (2) . Moreover, an -hyperideal of an ordered -semihypergroup is said to be completely prime if for each and such that , then or . An ordered -semihypergroup is called -regular if for every there exist and such that .
Theorem 5. Let be a nonempty subset of a commutative -regular ordered -semihypergroup . Then, the following assertions are equivalent:(1) is an -hyperfilter of .(2) or is a completely prime generalized -hyperideal of .
Proof. (1): Assume that (1) holds. Let . First of all, we show that is a generalized -hyperideal of , i.e., . Let . Since is -regular and , we getwhich is a contradiction. Suppose that and , where . If , then, since is an -hyperfilter of , we get , a contradiction. It implies that , and hence . Thus, is a generalized -hyperideal of . Now, we assert that is a completely prime generalized -hyperideal of . Let , , and . Then, there exists such that . If and , then, since is a sub -semihypergroup of , we get , a contradiction. Hence, or . Therefore, is a completely prime generalized -hyperideal of .(2): If , then , and hence is an -hyperfilter of . Let be a completely prime generalized -hyperideal of . Now, let and . If , then . Since is completely prime, we get or , which is a contradiction. Thus, and so is a sub -semihypergroup of . Now, let , , and . Let or . Without any loss of generality, we assume that . Since is -regular, we get . Then,Since is commutative and is an -hyperideal of , we haveIt implies that , which is a contradiction. Thus, and . Next, let and . If , then, since is an -hyperideal of , we get , a contradiction. So, , and thus is an -hyperfilter of .
4. Construction of Ordered -Semihypergroups
In this section, we consider some ordered -semihypergroups, where we define an ordered regular equivalence relation such that the quotient is an ordered -semihypergroup. More exactly, starting with an ordered -semihypergroup and using , we can construct an ordered -semihypergroup structure on the quotient set.
Example 3. LetDefine the hyperoperation and (partial) order relation on as follows:Then, is an ordered -semihypergroup. We give the covering relation asThe figure of is shown in Figure 3.
Here, it is a routine matter to verify that is an -hyperideal of .
Example 4. Suppose thatand . DefineWe consider the ordered -semihypergroup , where the (partial) order relation is defined by the following table:We give the covering relation asThe figure of is shown in Figure 4.
Suppose that . Clearly, is an -hyperfilter of for . Indeed: since or , or for any and for , we have that is an -hyperfilter of for .
But is not a -hyperfilter of . Indeed: since or , or for any and , we have that is not a -hyperfilter of .
Definition 6. Let be an ordered -semihypergroup. A relation on is called a weak pseudoorder on if(1).(2) and imply for all .(3) implies and for all and .(4) and imply and for all and .Clearly, every pseudoorder relation on an ordered -semihypergroup is a weak pseudoorder. The converse is not true, in general, that is, a weak pseudoorder may not be a pseudoorder of (see Example 6).
Theorem 6. Let be an ordered -semihypergroup and a weak pseudoorder on . Then, there exists a regular equivalence relationon such that is an ordered -semihypergroup, wheresuch that .
Proof. The proof is similar to the proof of Theorem 4 in [16].
Example 5. Letand . Define the hyperoperations and (partial) order relation on as follows:Then, is an ordered -semihypergroup. We give the covering relation asThe figure of is shown in Figure 5.
Here, is a -hyperfilter of . Let us consider the following relation in :Now, we consider a regular equivalence relation on as follows:By definition of , we getIf we takethen, by Theorem 6, is an ordered -semihypergroup, where and are defined bysuch that .
We give the covering relation asThe figure of is shown in Figure 6.
Example 6. Letand be the sets of binary hyperoperations defined as follows:We have that is an ordered -semihypergroup where the (partial) order relation is defined byWe give the covering relation asThe figure of is shown in Figure 7.
has no proper -hyperfilter. Now, we putClearly, is not a pseudoorder, since does not hold. Indeed:We haveThen, , where , , , , and . By Theorem 6, is an ordered -semihypergroup, where , , and are defined byWe give the covering relation asThe figure of is shown in Figure 8.
In the following example, we show that the converse of Lemma 1 is not true in general, i.e., an -hyperideal may not be an -hyperideal of an ordered -semihypergroup .
Example 7. We come back to Example 6 and consider ordered -semihypergroup . Put . Clearly, is a sub -semihypergroup of . We have  for all . . .Therefore, is an -hyperideal of but is not -hyperideal of .
5. Conclusions
One of the most important research areas in ordered -semihypergroup theory is the investigation of hyperfilters [25]. Generalization of hyperfilters in (ordered) -semihypergroups is necessary for further study of (ordered) -semihypergroups. In this paper, we studied some properties of -hyperideals and -hyperfilters of ordered -semihypergroups. For future work, it will be interesting to study relative -hyperideals and intuitionistic fuzzy -hyperideals in ordered hyperstructures such as ordered -semihypergroups, ordered Krasner hyperrings, and ordered semihyperrings. We expect further research efforts in this direction. These observations motivate the following future works: describe relations between fuzzy -hyperfilters and -fuzzy -hyperfilters.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this study.
Acknowledgments
This study was supported by the National Key R&D Program of China (no. 2018YFB1005100).