# Some Properties of WeakΓ-Hyperfilters in OrderedΓ-Semihypergroups

Authors: Yongsheng Rao; Xiang Chen; Saeed Kosari; Mohammadsadegh Monemrad
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The main purpose of this paper is to study fundamental properties of weakΓ-hyperfilters on ordered Γ-semihypergroups that is a generalization of Γ-hyperfilters. Also, we investigate the relationship between weak Γ-hyperfilters and prime Γ-hyperideals in ordered Γ-semihypergroups. Finally, we introduce weak m,n-Γ-hyperfilters of ordered Γ-semihypergroups and conclude in paper with arising explicit theorems.


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## Body

## 1. Introduction and Preliminaries

The theory of hypergroups was introduced by Marty [1] in 1934. For details about hyperstructure theory and its applications in cryptography, codes, etc, we refer to [2]. In the range of hyperstructures, Heidari and Davvaz have investigated the ordered semihypergroups [3]. In [4], Davvaz et al. provided a construction method for an ordered semigroup by using the pseudoorders. The study of ordered regular equivalence relations is one interesting topic in ordered semihypergroup theory. Gu and Tang [5] and Tang et al. [6] started the study of ordered regular equivalence relation in detail and achieved some results in this respect. The concept of hyperfilter is the most important subject in ordered semihypergroup. In [7], Mahboob and Khan introduced m,n-hyperfilters on ordered semihypergroups and studied its various properties and characterizations.Many researchers have studied (ordered)Γ-semihypergroups and their related notions, for instance, Omidi et al. [8, 9], Roa et al. [10], and Tang et al. [11], also see [12, 13]. In [9], Omidi et al. investigated the notion of (intra-) regular ordered Γ-semihypergroups associated with bi-Γ-hyperideals, and in [8], Omidi and Davvaz introduced the notion of convex ordered Γ-semihypergroups. In [14], Yaqoob and Tang investigated approximations of interior hyperfilters in partially ordered LA-semihypergroups. Recently, Rao et al. [15] studied some aspects of m-k-hyperideals in ordered semihyperrings.Fuzzy set theory has been developed in many directions. In 2015, the notion of fuzzy hyperfilters was introduced and studied by Tang et al. in [16]. In [17], Cheng and Xin investigated some types of hyperfilters on hyper-BE-algebras. In 2014, Borzooei et al. [18] studied weak filters in hyper-residuated lattices. For more results on derivations in ordered semihyperrings, one can see [15].Recently, Rao et al. [10] investigated some properties of m,n-Γ-hyperfilters in ordered Γ-semihypergroups. In continuity of this paper, we study weak m,n-Γ-hyperfilters of ordered Γ-semihypergroups. In [19], Bouaziz and Yaqoob studied hyperfilters of ordered LA-semihypergroups in the framework of rough sets. Recently, Tang et al. defined and analyzed in [20] the weak hyperfilters of ordered semihypergroups. We know that weak hyperfilters are generalizations of hyperfilters while ordered semihypergroups generalize ordered semigroups.In the following, some notions on orderedΓ-semihypergroups are reviewed to facilitate this study (see [10, 12], for more details and basic definitions). Let S be a nonempty set and ∗S be the family of all nonempty subsets of S. Then, a mapping γ:S×S⟶∗S is called a hyperoperation on S. A hypergroupoid is a set S together with a hyperoperation γ. If ∅≠A,B⊆S and x∈S, then AγB=∪a∈Ab∈Baγb, xγA=xγA, and Bγx=Bγx.We suppose that the hypergroupoidS,γ is equipped with the relation, aγbγc=aγbγc, which means that ∪u∈bγcaγu=∪v∈aγbvγc.For everya,b,c∈S, then S is called a semihypergroup.Consider a nonempty setS and a nonempty set Γ with the following properties: for all a,b,x,y∈S and all α,β∈Γ, we have(1)
aγb⊆S(2)
Ifa,b,x,y∈S such that a=x and b=y, then aγb=xγy(3)
xαaβb=xαaβbEveryγ∈Γ denotes the hyperoperation. Such a set S is said to be a Γ-semihypergroup. The reader may see [21, 22], for detailed discussion.Let∅≠A,B⊆S. We define AγB=∪aγb|a∈A,b∈B and(1)AΓB=∪γ∈ΓAγB.Definition 1.
(see [11]). An ordered Γ-semihypergroup is a triple S,Γ,≤ that(1)
S,Γ is a Γ-semihypergroup(2)
S,≤ is a (partially) ordered set(3)
For allu,v,x∈S and all γ∈Γ, u≤v implies uγx≺¯vγx and xγu≺¯xγv
If∅≠A,B⊆S, then A≺¯B⇔∀a∈A,∃b∈B;a≤b.
Let∅≠I⊆S. I is defined as follows: I≔x∈S|x≤zfor somez∈I.
For convenience, givena∈I, we write a=a.
By a sub-Γ-semihypergroup of an ordered Γ-semihypergroup S, we mean a nonempty subset A of S such that AΓA⊆A, i.e., aγb⊆A, for every a,b∈A and γ∈Γ.Definition 2.
(see [12]). A nonempty subset I of an ordered Γ-semihypergroup S is called a Γ-hyperideal of S if(1)
SΓI⊆I and IΓS⊆I(2)
I=I
For any∅≠H⊆S, we define H by x∈S|h≤xfor someh∈H. For H=h, we write h instead of h.
The concept of anm,n-Γ-hyperfilter is a generalization of the concept of a Γ-hyperfilter of S. Also, see [10], for an overview.Definition 3.
(see [10]). If S,Γ,≤ is an ordered Γ-semihypergroup, then a left m-Γ-hyperfilter F is a sub-Γ-semihypergroup equipped with F⊆F and, in addition, for all a,b∈S and γ∈Γ,aγb∩F≠∅ implies am⊆F. Right n-Γ-hyperfilters can be defined similarly. Here, mandn are positive integers. If F is both a left m-Γ-hyperfilter and a right n-Γ-hyperfilter of S, then F is called an m,n-Γ-hyperfilter of S. For m=n=1, F is a Γ-hyperfilter of S.


## 2. Main Results

In this section, we develop several definitions and results on an orderedΓ-semihypergroup S,Γ,≤. Some remarkable properties associated with weak Γ-hyperfilters were investigated. Also, we examine the relationship between weak Γ-hyperfilters and prime Γ-hyperideals by mentioning a theorem.Definition 4.
LetFw≠∅ be a subset of an ordered Γ-semihypergroup S,Γ,≤. Then, Fw is called a left (right) weak Γ-hyperfilter of S if(1)
xγy∩Fw≠∅ for all x,y∈Fw and all γ∈Γ(2)
For allx,y∈S and all γ∈Γ,xγy∩Fw≠∅⟹x∈Fwy∈Fw(3)
For allx∈Fw and z∈S,x≤z⟹z∈Fw, i.e., Fw⊆Fw
Note that ifFw is both a left weak Γ-hyperfilter and a right weak Γ-hyperfilter of S, then Fw is called a weak Γ-hyperfilter of S.
Clearly, everyΓ-hyperfilter (see Definition 3) of an ordered Γ-semihypergroup S is a weak Γ-hyperfilter of S. The converse is not true, in general, that is, a weak Γ-hyperfilter may not be a Γ-hyperfilter of S.Example 1.
LetS=a,b,c,d and Γ=γ,β. The hyperoperations γ and β are given in Table 1 and Table 2. Then, S,Γ is a Γ-semihypergroup [23].
Now, we consider the (partial) order relation,(2)≤≔a,a,b,b,c,c,d,a,d,c,d,d,on S. Then, S,Γ,≤ is an ordered Γ-semihypergroup. Cover relation of S as given below:(3)≺=d,a,d,c.
The Hasse diagram ofS is shown in Figure 1.
Here,Fw=a,b,c is a weak Γ-hyperfilter of S. Clearly, Fw is not a Γ-hyperfilter of S. Since FwΓFw=S⊈Fw, i.e., aΓb=b,d⊈Fw, it follows that Fw is not a sub-Γ-semihypergroup of S.
Note that ifFwi|i∈I is a family of weak Γ-hyperfilters of an ordered Γ-semihypergroup S, for all i∈I, then ∪i∈IFwi is not a weak Γ-hyperfilter of S in general (see Example 3.3 in [20]). In the following, we show that if Fwi|i∈I is a chain of weak Γ-hyperfilters of S, then ∪i∈IFwi is a weak Γ-hyperfilter of S.Table 1
Table ofγ for Example 1.
γabcdaa{b, d}cdb{b,d}b{b,d}dcc{b,d}addddddTable 2
Table ofβ for Example 1.
βabcda{a, c}{b, d}{a, c}db{b, d}b{b, d}dc{a, c}{b, d}{a, c}ddddddFigure 1
Figure ofS for Example 1.Lemma 1.
Suppose thatFw1 and Fw2 are weak Γ-hyperfilters of an ordered Γ-semihypergroup S,Γ,≤. Then, Fw1∪Fw2 is a weak Γ-hyperfilter of S if and only if Fw1⊆Fw2 or Fw2⊆Fw1.Proof.
IfFw1⊆Fw2 or Fw2⊆Fw1, then it is clear that Fw1∪Fw2 is a weak Γ-hyperfilter of S.
Assume thatFw1∪Fw2 is a weak Γ-hyperfilter of S and Fw1⊈Fw2 and Fw2⊈Fw1. Then, there exist x,y∈Fw1∪Fw2 such that x∈Fw1, x∉Fw2, y∈Fw2, and y∉Fw1. Since Fw1∪Fw2 is a weak Γ-hyperfilter of S, we obtain xγy∩Fw1∪Fw2≠∅, for each γ∈Γ.Case 1.
xγy∩Fw1≠∅. Since Fw1 is a weak Γ-hyperfilter of S, we get y∈Fw1, a contradiction.Case 2.
Next, we consider the case whenxγy∩Fw2≠∅. Since Fw2 is a weak Γ-hyperfilter of S, we have x∈Fw2, a contradiction.
From Lemma1, we get the following result.Theorem 1.
SupposeFwi|i∈I is a family of weak Γ-hyperfilters of an ordered Γ-semihypergroup S,Γ,≤ such that Fwu⊆Fwv or Fwv⊆Fwu, for all u,v∈I. Then, ∪i∈IFwi is a weak Γ-hyperfilter of S, where ∣I∣≥2.Proof.
Straightforward.Theorem 2.
LetS,Γ,≤ be an ordered Γ-semihypergroup and ∅≠Fw⊆S. Then, the following statements are equivalent:(1)
Fw is a weak Γ-hyperfilter of S(2)
S∖Fw=∅ or S∖Fw is a prime Γ-hyperideal of SProof.
1⟹2: assume that (1) holds and S∖Fw≠∅. We first show that S∖Fw is a Γ-hyperideal of S. Let a∈S, b∈S∖Fw, and γ∈Γ. If bγa⊈S∖Fw, then there exists x∈bγa such that x∈Fw. So, bγa∩Fw≠∅. Since Fw is a weak Γ-hyperfilter of S, we get b∈Fw, which is a contradiction. So, bγa⊆S∖Fw. It means that S∖FwΓS⊆S∖Fw.
Similarly,SΓS∖Fw⊆S∖Fw.
Now, letb∈S∖Fw, a∈S, and a≤b. Since Fw is a weak Γ-hyperfilter of S, it follows that Fw⊆Fw. If a∈Fw, then b∈Fw, which is a contradiction. It implies that a∈S∖Fw. Hence, S∖Fw⊆S∖Fw. Hence, by Definition 2, S∖Fw is a Γ-hyperideal of S.
Next, we show thatS∖Fw is prime. Let a,b∈S, γ∈Γ, and aγb⊆S∖Fw. If a∈Fw and b∈Fw, then, since Fw is a weak Γ-hyperfilter of S, we get aγb∩Fw≠∅, a contradiction. So, a∈S∖Fw or b∈S∖Fw. Therefore, S∖Fw is a prime Γ-hyperideal of S.
2⟹1: if S∖Fw=∅, then Fw=S, and so Fw is a weak Γ-hyperfilter of S. Now, let S∖Fw is a prime Γ-hyperideal of S. We assert that Fw is a weak Γ-hyperfilter of S. Let a,b∈Fw and γ∈Γ. If aγb∩Fw=∅, then aγb⊆S∖Fw. Since S∖Fw is prime, it follows that a∈S∖Fw or b∈S∖Fw, which is a contradiction. Hence, aγb∩Fw≠∅, for all a,b∈Fw and γ∈Γ. So, the first condition of Definition 4 is verified.
Now, leta,b∈S, γ∈Γ, and aγb∩Fw≠∅. If a∈S∖Fw, then aγb⊆S∖FwΓS Since S∖Fw is a Γ-hyperideal, we get aγb⊆S∖Fw. So, aγb∩Fw≠∅, which is a contradiction. It implies that a∈Fw. Similarly, b∈Fw. Thus, the second condition of Definition 4 is verified.
Assume thata∈Fw and a≤x, where x∈S. If x∈S∖Fw, then, since S∖Fw is a Γ-hyperideal of S, we get a∈S∖Fw, a contradiction. So, x∈Fw, and hence, the third condition of Definition 4 is verified. Therefore, Fw is a weak Γ-hyperfilter of S.Theorem 3.
Suppose thatW is a weak Γ-hyperfilter of a commutative ordered Γ-semihypergroup S,Γ,≤. If W≪aγb and a∈W, then b∈W for all a,b∈S and γ∈Γ. Here, U≪V means that there exist u∈U and v∈V such that u≤v, for all nonempty subsets U and V of S.Proof.
LetW be a weak Γ-hyperfilter of S, a∈W and W≪aγb, where b∈S. As W≪aγb, there exists u∈W and v∈aγb such that u≤v. Since W is a weak Γ-hyperfilter of S, we get v∈W. It implies that aγb∩W≠∅, for each γ∈Γ. By condition (2) of Definition 4, we obtain b∈W.Theorem 4.
LetW be a weak Γ-hyperfilter of an ordered Γ-semihypergroup S,Γ,≤. If U∩W≠∅ and U≺¯V, then V∩W≠∅, where ∅≠U,V⊆S.Proof.
SinceU∩W≠∅, then there exists x∈S such that x∈W and x∈U. As U≺¯V and x∈U, there exists y∈V such that x≤y. Since W is a weak Γ-hyperfilter of S and x∈W, we have y∈W, by condition (3) of Definition 4. So, V∩W≠∅.
A proper weakΓ-hyperfilter Fw of S is said to be maximal if there does not exist any proper weak Γ-hyperfilter of S which properly contains Fw. Indeed, let Fw be a proper weak Γ-hyperfilter of S. Then, Fw is said to be maximal if Fw⊆W⊆S implies Fw=W or W=S, for all weak Γ-hyperfilters W of S.Example 2.
In Example1, Fw=a,b,c is a maximal weak Γ-hyperfilter of S.Theorem 5.
LetS,Γ,≤ be an ordered Γ-semihypergroup with zero. If Fw is a proper weak Γ-hyperfilter of S, then there is a maximal weak Γ-hyperfilter of S containing Fw.Proof.
SetX=W⊂≠S|W as a weak Γ-hyperfilter of S, Fw≠S and Fw⊆W.
SinceFw∈X, we have X≠∅. Also, X,⊆ is an ordered set by inclusion relation ⊆. Let Wi|i∈I be a chain in X. By Theorem 1, ∪i∈IWi is a weak Γ-hyperfilter of S. Clearly, ∪i∈IWi is an upper bound of the chain Wi|i∈I. Since 0∉Wi for any i∈I, we get 0∉∪i∈IWi. This shows that ∪i∈IWi⊂≠S. So, ∪i∈IWi∈X. By Zorn’s Lemma, there exists a maximal element Mw in X. We assert that Mw is a maximal weak Γ-hyperfilter of S. Now, let J be a proper weak Γ-hyperfilter of S containing Mw. Then, J contains Fw and so it belongs to X. Since Mw is maximal in X, we get J=Mw. Therefore, Mw is a maximal weak Γ-hyperfilter of S.
The weakm,n-Γ-hyperfilter of ordered Γ-semihypergroups is defined as follows.Definition 5.
LetFw≠∅ be a subset of an ordered Γ-semihypergroup S,Γ,≤. Then, Fw is said to be a left weak m-Γ-hyperfilter (right weak n-Γ-hyperfilter) of S if(1)
For alla,b∈Fw and γ∈Γ,aγb∩Fw≠∅(2)
For alla,b∈S and γ∈Γ,aγb∩Fw≠∅⟹am⊆Fwbn⊆Fw(3)
For alla∈Fw and z∈S,a≤z⟹z∈Fw
Here,mandn are positive integers. Note that if Fw is both a left weak m-Γ-hyperfilter and a right weak n-Γ-hyperfilter of S, then Fw is called a weak m,n-Γ-hyperfilter of S.
Clearly, everym,n-Γ-hyperfilter of S is a weak m,n-Γ-hyperfilter. However, the converse is not true, in general, that is, a weak m,n-Γ-hyperfilter may not be an m,n-Γ-hyperfilter of S.Example 3.
SupposeS is Γ-semihypergroup in Example 1 and put(4)≤≔a,a,b,b,c,c,d,b,d,d.
Then,S,Γ,≤ is an ordered Γ-semihypergroup. The covering relation of S is given by(5)≺=d,b.
The figure ofS is shown in Figure 2.
Clearly,Fw=a,b,c is a weak m,n-Γ-hyperfilter on S, but it is not an m,n-Γ-hyperfilter. Indeed, a,b∈FwandaΓb=b,d⊈Fw.Figure 2
Figure ofS for Example 3.Lemma 2.
LetFwi|i∈I be a family of weak m,n-Γ-hyperfilters of an ordered Γ-semihypergroup S,Γ,≤. If ∩i∈IFwi≠∅, then ∩i∈IFwi is a weak m,n-Γ-hyperfilter of S.Proof.
Letx,y∈∩i∈IFwi. Then, x,y∈Fwi, for each i∈I. Since Fwi is a weak m,n-Γ-hyperfilter of S for each i∈I, we get xγy∩Fwi≠∅, for all γ∈Γ. It implies that xγy∩∩i∈IFwi≠∅.
Now, letx,y∈S, γ∈Γ, and xγy∩∩i∈IFwi≠∅. Then, there exists u∈∩i∈IFwi, for some u∈xγy. Since u∈∩i∈IFwi, it follows that u∈Fwi, for each i∈I. Since Fwi is a weak m,n-Γ-hyperfilter of S, for each i∈I, we get xm,yn⊆Fwi for each i∈I. It implies that xm,yn⊆∩i∈IFwi.
Now, letx∈∩i∈IFwi and x≤z∈S. Then, x∈Fwi for each i∈I. Since Fwi is a weak m,n-Γ-hyperfilter of S, for all i∈I, we have z∈Fwi, for all i∈I. So, z∈∩i∈IFwi. Therefore, ∩i∈IFwi is a weak m,n-Γ-hyper filter of S.
LetSi,Γi,≤i be an ordered Γi-semihypergroup, for all i∈Ω. Define ⊙:∏i∈ISi×∏i∈IΓi×∏i∈ISi⟶∗∏i∈ISi by uii∈Ω⊙γii∈Ω⊙vii∈Ω=tii∈Ω|ti∈uiγivi, for all uii∈Ω,vii∈Ω∈∏i∈ΩSi and γii∈Ω∈∏i∈ΩΓi. Set uii∈Ω≤vii∈Ω if and only if, for all i∈Ω, ui≤ivi.
Then,∏i∈ΩSi=uii∈Ω|ui∈Si,∏i∈ΩΓi,≤ is an ordered ∏i∈ΩΓi-semihypergroup [8]. In the following, we study the behavior of weak m,n-Γ-hyperfilters on ∏i∈ΩSi.Theorem 6.
LetSi,Γi,≤i be an ordered Γi-semihypergroup, for all i∈Ω. If Wi is a weak m,n-Γ-hyperfilter on Si, for all i∈Ω, then F=∏i∈ΩWi is a weak m,n-Γ-hyperfilter on ∏i∈ΩSi.Proof.
Letuii∈Ω,vii∈Ω∈F=∏i∈ΩWi. Then, ui,vi∈Wi, for each i∈Ω. As Wi is a weak m,n-Γ-hyperfilter of Si, we have uiγivi∩Wi≠∅. So, uii∈Ω⊙γii∈Ω⊙vii∈Ω=uiγivii∈Ω∩F≠∅.
Now, letuii∈Ω,vii∈Ω∈∏i∈ΩSi and uii∈Ω⊙γii∈Ω⊙vii∈Ω∩F≠∅. Then,(6)uii∈Ω⊙γii∈Ω⊙vii∈Ω∩F≠∅⟹uiγivii∈Ω∩F≠∅⟹uiγivi∩Wi≠∅,∀i∈Ω⟹uim⊆Wiandvin⊆Wi,∀i∈Ω⟹uimi∈Ω⊆Fandvini∈Ω⊆F⟹uii∈Ωm⊆Fandvii∈Ωn⊆F.
Letuii∈Ω∈F and aii∈Ω∈∏i∈ΩSi such that uii∈Ω,aii∈Ω∈≤. Then, for all i∈Ω, we have ui,ai∈≤i. Since Wi is a weak m,n-Γ-hyperfilter of Si, for each i∈Ω, we have ai∈Wi, for each i∈Ω. It implies that aii∈Ω∈∏i∈ΩWi=F. Therefore, F is a weak m,n-Γ-hyperfilter of ∏i∈ΩSi.


## 3. Conclusions

Generalization ofΓ-hyperfilters in ordered Γ-semihypergroups is necessary for further study of ordered Γ-semihypergroups. In this study, we introduced the notion of weak m,n-Γ-hyperfilter and then obtained some related basic results. In the future, we plan to study relative weak Γ-hyperfilters, fuzzy weak Γ-hyperfilters, and rough weak Γ-hyperfilters in ordered Γ-semihypergroups. We expect further research efforts in this direction.Question 1: under what condition a weakm,n-Γ-hyperfilter of S coincides with a weak Γ-hyperfilter?Question 2: under what condition arbitrary union of weakm,n-Γ-hyperfilters of S is a weak m,n-Γ-hyperfilter?



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