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