# A Study onA−I−Γ-Hyperideals and m,n−Γ-Hyperfilters in Ordered Γ-Semihypergroups

Authors: Yongsheng Rao; Saeed Kosari; Zehui Shao; Maryam Akhoundi; Saber Omidi
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

The concept of almost interiorΓ-hyperideals (A−I−Γ-hyperideals) in ordered Γ-semihypergroups is a generalization of the concept of interior Γ-hyperideals (I−Γ-hyperideals). In this study, the connections between I−Γ-hyperideals and A−I−Γ-hyperideals in ordered Γ-semihypergroups were presented. Also, we define the notion of m,n-Γ-hyperfilters of ordered Γ-semihypergroups and provide useful results on it. Moreover, we use the weak pseudoorders to construct quotient ordered Γ-semihypergroups by using ordered regular equivalence relations. These concepts lead us to a new research direction in ordered Γ-semihypergroups.


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## Body

## 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 (Hν-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 Γ-semihypergroups and discussed the Γ-hyperideals of Γ-semihypergroups. In [18], the concept of (fuzzy) Γ-hyperideals of involution Γ-semihypergroups was investigated. Recently, fuzzy set theory has been well developed in the framework of ordered Γ-semihypergroup theory [19]. In 2020, Rao et al. [20] introduced and studied the concept of relative bi-(int-)Γ-hyperideals in ordered Γ-semihypergroups. The beauty of relative bi-(int-)Γ-hyperideals is that instead of involving all the elements of the ordered Γ-semihypergroup S, we deal with the elements belonging to the nonempty subsets of S. The concept of A-ideals of semigroups was introduced by Grosek and Satko [21] and was further investigated in [22]. In [23], Suebsung et al. studied A-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 Γ-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 m,n-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 m,n-hyperfilters to ordered Γ-semihypergroups. In Section 2, we give the preliminary concepts concerning ordered Γ-semihypergroups and I−Γ-hyperideals. In Section 3, the notion of almost interior Γ-hyperideals (briefly, A−I−Γ-hyperideal) is given. In addition, some properties of m,n−Γ-hyperfilters in ordered Γ-semihypergroups have been proven. Finally, we conclude in Section 4 with some arising examples. Theory of A−I−Γ-hyperideals and m,n−Γ-hyperfilters is useful to explore new results associated with ordered Γ-semihypergroups.


## 2. Preliminaries

LetS be a nonempty set and P∗S be the family of all nonempty subsets of S. A mapping γ:S×S⟶P∗S is called a hyperoperation on S. A hypergroupoid is a set S together with a (binary) hyperoperation. If A and B are two nonempty subsets of S and x∈S, then we denote(1)AγB=∪a∈Ab∈Baγb,xγA=xγA,Bγx=Bγx.A hypergroupoidS,γ is called a semihypergroup if for every a,b,c in S,(2)aγbγc=aγbγc.The notion ofΓ-semihypergroups has been introduced by Anvariyeh et al. [17] in 2010. In this section, we present some notions related to (ordered) Γ-semihypergroup theory.Definition 1 (see [17]).
LetS and Γ be two nonempty sets. Then, S is said to be a Γ-semihypergroup if every γ∈Γ is a hyperoperation on S, i.e., xγy⊆S for every x,y∈S, and for every α,β∈Γ and x,y,z∈S, we have(3)xαyβz=xαyβz.LetA and B be two nonempty subsets of S. We define(4)AγB=∪aγb|a∈A,b∈B.Also,(5)AΓB=∪aγb|a∈A,b∈Bandγ∈Γ=∪γ∈ΓAγB.If everyγ∈Γ is an operation, then S is a Γ-semigroup. A nonempty subset A of S is called a sub Γ-semihypergroup of S if xβy⊆A for every x,y∈A and β∈Γ.Definition 2.
(see [25]). Let ≤ be an ordered relation on a nonempty set S. By an ordered Γ-semihypergroup we mean an algebraic hyperstructure S,Γ,≤ in which the following are satisfied:(1)
S,Γ is a Γ-semihypergroup.(2)
S,≤ is a (partially) ordered set.(3)
For anya,x,y∈S and β∈Γ, x≤y implies aβx⪯¯aβy and xβa⪯¯yβa. If U and V are nonempty subsets of S, then(6)U⪯¯V⟺∀u∈U,∃v∈V;u≤v.
Letσ be an equivalence relation on an ordered Γ-semihypergroup S,Γ,≤. If U and V are nonempty subsets of S, then(1)
Uσ⟶V⟺∀u∈U,∃v∈V;uσv.(2)
Uσ←V⟺∀v′∈V,∃u′∈U;u′σv′.(3)
Uσ˜V⟺∀u∈U,∃v∈V;uσv and vσu&∀v′∈V,∃u′∈U;v′σu′ and u′σv′.(4)
Uσ¯V⟺Uσ⟶V and Uσ←V.(5)
Uσ¯¯V⟺∀u∈U,∀v∈V,uσv.
A relationσ on an ordered Γ-semihypergroup S,Γ,≤ is called a pseudoorder [25] on S if (1) ≤⊆σ; (2) aσb and bσc imply aσc for all a,b,c∈S; (3) aσb implies aγcσ¯¯bγc and cγaσ¯¯cγb for all a,b,c∈S and γ∈Γ.Theorem 1 (see [25]).
LetS,Γ,≤ be an ordered Γ-semihypergroup and σ a pseudoorder on S. Then, there exists a strongly regular equivalence relation σ∗=a,b∈S×S|aσbandbσa on S such that S/σ∗,Γσ∗,⪯¯σ∗ is an ordered Γσ∗-semigroup, where ⪯¯σ∗≔σ∗x,σ∗y∈S/σ∗×S/σ∗|∃a∈σ∗x,∃b∈σ∗ysuch thata,b∈σ.LetH be a nonempty subset of an ordered Γ-semihypergroup S,Γ,≤. H is defined as follows:(7)H≔x∈S|x≤hfor someh∈H.For convenience, givenh∈H, we write h=h. Let A and B be nonempty subsets of an ordered Γ-semihypergroup S,Γ,≤. Then,(1)
A⊆A.(2)
IfA⊆B, then A⊆B.(3)
AΓB⊆AΓB and AΓB=AΓB.(4)
A=A.LetS be an ordered Γ-semihypergroup. By a sub Γ-semihypergroup of S we mean a nonempty subset A of S such that AΓA⊆A. We denote(8)An=AΓAΓ⋯ΓA︸n−copies.The interiorΓ-hyperideal (in short I−Γ-hyperideal) is defined as follows.Definition 3 (see [19]).
LetS,Γ,≤ be an ordered Γ-semihypergroup. A sub Γ-semihypergroup A of S is called an interior Γ-hyperideal (in short I−Γ-hyperideal) of S if(1)
SΓAΓS⊆A.(2)
A⊆A.An orderedΓ-semihypergroup S,Γ,≤ is called regular [31] if for every a∈S there exist x∈S, α,β∈Γ such that a≤aαxβa. This is equivalent to saying that a∈aΓSΓa, for every a∈S or A⊆AΓSΓA, for every A⊆S.


## 3.A−I−Γ-Hyperideals and m,n−Γ-Hyperfilters

Throughout the rest of this paper:S will be an ordered Γ-semihypergroup. We begin this section with the definition of an almost interior Γ-hyperideal (in short A−I−Γ-hyperideal) on an ordered Γ-semihypergroup S,Γ,≤.Definition 4.
LetS,Γ,≤ be an ordered Γ-semihypergroup. A nonempty subset K of S is called an A−I−Γ-hyperideal of S if(1)
xΓKΓy∩K≠∅ for any x,y∈S.(2)
K⊆K.Example 1.
LetS=a,b,c,d and Γ=γ,β. Define hyperoperations γ and β on S by the following tables:(9)
Then,S,Γ is a Γ-semihypergroup [32]. We have that S,Γ,≤ is an ordered Γ-semihypergroup where the (partial) order relation ≤ is defined by(10)≤≔a,a,b,b,c,c,d,a,d,c,d,d.
We give the covering relation≺ as(11)≺=d,a,d,c.
The Hasse diagram ofS is shown in Figure 1.
Here,I=a,c,d is an A−I−Γ-hyperideal of S.Figure 1
Figure ofS for Example 1.Lemma 1.
LetS,Γ,≤ be an ordered Γ-semihypergroup. Then, every I−Γ-hyperideal of S is an A−I−Γ-hyperideal of S.Proof.
LetK be an I−Γ-hyperideal of S. Then, SΓKΓS⊆K. If a,b∈S, then(12)aΓKΓb⊆SΓKΓS⊆K.
It implies thataΓKΓb⊆K⊆K. So, aΓKΓb∩K≠∅ for any a,b∈S. Therefore, K is an A−I−Γ-hyperideal of S.In Section4, we show that the converse of Lemma 1 is not true in general, i.e., an A−I−Γ-hyperideal may not be an I−Γ-hyperideal of S.Lemma 2.
LetS,Γ,≤ be an ordered Γ-semihypergroup. If K1 and K2 are A−I−Γ-hyperideals of S, then K1∪K2 is an A-I-Gamma-hyperideal of S.Proof.
Clearly,K1⊆K1∪K2. If a,b∈S, then(13)aΓK1Γb⊆aΓK1∪K2Γb.
So,aΓK1Γb⊆aΓK1∪K2Γb. It implies that(14)aΓK1Γb∩K1⊆aΓK1∪K2Γb∩K1∪K2.
SinceK1 is an A−I−Γ-hyperideal of S, it follows that aΓK1Γb∩K1≠∅. This means that aΓK1∪K2Γb∩K1∪K2≠∅. Therefore, K1∪K2 is an A−I−Γ-hyperideal of S.Theorem 2.
LetS,Γ,≤ be an ordered Γ-semihypergroup and S>1. Then, the following assertions are equivalent:(1)
S has no proper A−I−Γ-hyperideal.(2)
For anyx∈S, there exists ax∈S such that axΓS\xΓax=x.Proof.
(1)
⇒2: Assume that (1) holds. Let x∈S. Then, S\x is not a A−I−Γ-hyperideal. So, there exists ax∈S such that axΓS\xΓax∩S\x=∅. It implies that axΓS\xΓax=x.(2)
⇒1: Suppose that for any x∈S, there exists ax∈S such that axΓS\xΓax=x. This gives(15)axΓS\xΓax∩S\x=x∩S\x=∅.
Therefore,S\x is not an A−I−Γ-hyperideal for every x∈S. Now, let K be a proper A−I−Γ-hyperideal of S. Then, K⊆S\x for some x∈S. Thus, xΓKΓx⊆xΓS\xΓx, and hence xΓKΓx∩K⊆xΓS\xΓx∩S\x. Since K is an A−I−Γ-hyperideal of S, it follows that ∅≠xΓKΓx∩K. So, ∅≠xΓS\xΓx∩S\x. This means that S\x is an A−I−Γ-hyperideal of S, which is a contradiction. This completes the proof.In the following, as a continuation of the previous study [25], the authors discuss m,n−Γ-hyperfilter in ordered Γ-semihypergroups. The m,n−Γ-hyperfilter of ordered Γ-semihypergroups is defined as follows.Definition 5.
A subΓ-semihypergroup F of an ordered Γ-semihypergroup S,Γ,≤ is called a left m−Γ-hyperfilter (resp. right n−Γ-hyperfilter) of S if(1)
For alla,b∈S and γ∈Γ,aγb∩F≠∅⇒am⊆F (resp. bn⊆F).(2)
For alla∈F and c∈S,a≤c⇒c∈F.
Here,m,n are positive integers. Note that 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.
LetK be a sub Γ-semihypergroup of an ordered Γ-semihypergroup S,Γ,≤. We define(16)K≔x∈S|x≥kfor somek∈K.
ForK=k, we write k instead of k. Note that condition (2) in Definition 5 is equivalent to F⊆F.
The concept of anm,n−Γ-hyperfilter is a generalization of the concept of a Γ-hyperfilter of S. An m,n−Γ-hyperfilter of S is a k,l−Γ-hyperfilter of S for all k≥m and l≥n. For m=1=n, F is a Γ-hyperfilter of S. The following example shows that any m,n−Γ-hyperfilter of an ordered Γ-semihypergroup S need not always be a Γ-hyperfilter of S. Also, see Example 4 in Section 4.Example 2.
LetS=a,b,c,d and Γ=γ. We define(17)
Then,S is a Γ-semihypergroup. We have that S,Γ,≤ is an ordered Γ-semihypergroup where the order relation ≤ is defined by(18)≤≔a,a,a,b,a,c,b,b,c,c,d,d.
We give the covering relation≺ as(19)≺=a,b,a,c.
The figure ofS is shown in Figure 2.
Letℱ=a,b,c. Clearly, ℱ is an m,n−Γ-hyperfilter of S for m,n≥2. But ℱ is not a Γ-hyperfilter of S. Indeed: since dγx=a or a,b⊆ℱ, xγd=a or a,b⊆ℱ for any x∈S, dm⊆ℱ for m≥2 and d∉ℱ, we have that ℱ is an m,n−Γ-hyperfilter of S for m,n≥2 but not a Γ-hyperfilter of S.Figure 2
Figure ofS for Example 2.Lemma 3.
LetFλ|λ∈Λ be a family of m,n−Γ-hyperfilters of an ordered Γ-semihypergroup S,Γ,≤. If ∩λ∈ΛFλ≠∅, then ∩λ∈ΛFλ is an m,n−Γ-hyperfilter of S.Proof.
Leta,b∈∩λ∈ΛFλ. Then, a,b∈Fλ for each λ∈Λ. Since Fλ is a sub Γ-semihypergroup of S for each λ∈Λ, it follows that aγb⊆∩λ∈ΛFk for all γ∈Γ. Hence, ∩λ∈ΛFλ is a sub Γ-semihypergroup of S. Now, let a,b∈S, γ∈Γ and aγb∩∩λ∈ΛFλ≠∅. Then, there exists x∈∩λ∈ΛFλ for some x∈aγb. Since x∈∩λ∈ΛFλ, it follows that x∈Fλ for each λ∈Λ. Since Fλ is an m,n−Γ-hyperfilter of S for each λ∈Λ, we get xm,yn⊆Fλ for each λ∈Λ. It implies that xm,yn⊆∩λ∈ΛFλ. Now, let a∈∩λ∈ΛFλ and a≤c∈S. Then, a∈Fλ for each λ∈Λ. Since Fλ is an m,n−Γ-hyperfilter of S for all λ∈Λ, it follows that c∈Fλ for all λ∈Λ. So, c∈∩λ∈ΛFλ. Therefore, ∩λ∈ΛFλ is an m,n−Γ-hyperfilter of S.
One of the distinguished properties of them,n−Γ-hyperfilters is that their union is not an m,n−Γ-hyperfilter in general, for example, see [25, 30].
LetSi,Γi,≤i be an ordered Γi-semihypergroup for all i∈Ω. Define ⊙:∏i∈ISi×∏i∈IΓi×∏i∈ISi⟶P∗∏i∈ISi by(20)aii∈Ω⊙γii∈Ω⊙bii∈Ω=cii∈Ω|ci∈aiγibi,for all aii∈Ω,bii∈Ω∈∏i∈ΩSi, and γii∈Ω∈∏i∈ΩΓi. Now, we put aii∈Ω≤bii∈Ω if and only if, for all i∈Ω, ai≤ibi.
Then,∏i∈ΩSi=aii∈Ω|ai∈Si,∏i∈ΩΓi,≤ is an ordered ∏i∈ΩΓi-semihypergroup [33]. In the following, we study the behavior of m,n−Γ-hyperfilters on the product of ordered Γ-semihypergroups.Theorem 3.
LetFi be an m,n−Γ-hyperfilter on the ordered Γi-semihypergroup Si,Γi,≤i for all i∈Ω. Then, F=∏i∈ΩFi is an m,n−Γ-hyperfilter on ∏i∈ΩSi.Proof.
We divide the proof into three steps.Step 1. We first show that F is a sub Γ-semihypergroup of ∏i∈ΩSi.Letaii∈Ω,bii∈Ω∈F=∏i∈ΩFi. Then, ai,bi∈Fi for each i∈Ω. As Fi’s is a sub Γ-semihypergroup of Si, we have aiγibi⊆Fi. So, we have(21)aii∈Ω⊙γii∈Ω⊙bii∈Ω=aiγibii∈Ω∈∏i∈ΩFi=F.Thus,F is a sub Γ-semihypergroup of ∏i∈ΩSi.Step 2. Now, let aii∈Ω,bii∈Ω∈∏i∈ΩSi and aii∈Ω⊙γii∈Ω⊙bii∈Ω∩F≠∅. Then,(22)aii∈Ω⊙γii∈Ω⊙bii∈Ω∩F≠∅⇒aiγibii∈Ω∩F≠∅⇒aiγibi∩Fi≠∅,∀i∈Ω⇒aim⊆Fi,bin⊆Fi,∀i∈Ω⇒aimi∈Ω⊆F,bini∈Ω⊆F⇒aii∈Ωm⊆F,bii∈Ωn⊆F.Step 3. Let aii∈Ω∈F and cii∈Ω∈∏i∈ΩSi such that aii∈Ω,cii∈Ω∈≤. Then, for all i∈Ω, we have ai,ci∈≤i. Since Fi is an m,n−Γ-hyperfilter of Si for each i∈Ω, we get ci∈Fi for each i∈Ω. It implies that cii∈Ω∈∏i∈ΩFi=F. Therefore, F is an m,n−Γ-hyperfilter of ∏i∈ΩSi.A mappingφ:S⟶T of an ordered Γ-semihypergroup S,Γ,≤S into an ordered Γ-semihypergroup T,Γ′,≤T is said to be a normal Γ-homomorphism if (1) φaγb=φaγ′φb for all a,b∈S, γ∈Γ and γ′∈Γ′; (2) φ is isotone, i.e., for any a,b∈S, a,b∈≤S implies φa,φb∈≤T.Theorem 4.
Letφ:S⟶T be a normal Γ-homomorphism of ordered Γ-semihypergroups S,Γ,≤S and T,Γ′,≤T. If F is an m,n−Γ-hyperfilter of T, then(23)φ−1F=x∈S|φx∈F,is an m,n−Γ-hyperfilter of S.Proof.
We divide the proof into three steps.Step 1. Let x,y∈φ−1F, γ∈Γ, and γ′∈Γ′. Then, φx,φy∈F. Since F is a sub Γ-semihypergroup of T and φ is normal Γ-homomorphism, we have(24)φxγy=φxγ′φy⊆FΓF⊆F.It implies thatxγy⊆φ−1F. Thus, φ−1F is sub Γ-semihypergroup of S.Step 2. Let x,y∈S, γ∈Γ, and xγy∩φ−1F≠∅. Then,(25)xγy∩φ−1F≠∅⇒φxγy∩F≠∅⇒φxγ′φy∩F≠∅⇒φxm⊆F,φyn⊆F⇒φxγ′φxγ′⋯γ′φx︸m−copies⊆F,φyγ′φyγ′⋯γ′φy︸n−copies⊆F⇒φxγ′xγ′⋯γ′x︸m−copies⊆F,φyγ′yγ′⋯γ′y︸n−copies⊆F⇒φxm⊆F,φyn⊆F⇒xm⊆φ−1F,yn⊆φ−1F.Step 3. Now, suppose that a∈φ−1F and c∈S such that a≤Sc. Then, φa∈F. Since a≤Sc and φ is normal Γ-homomorphism, we have φa≤Tφc. Since F is an m,n−Γ-hyperfilter of T, we get φc∈F. So, c∈φ−1F. Therefore, φ−1F is an m,n−Γ-hyperfilter of S.LetK be a sub Γ-semihypergroup (resp. nonempty subset) of an ordered Γ-semihypergroup S,Γ,≤. Then, K is called an m,n−Γ-hyperideal (resp. generalized m,n−Γ-hyperideal) [34] of S if (1) KmΓSΓKn⊆K and (2) K⊆K. Moreover, an m,n−Γ-hyperideal K of an ordered Γ-semihypergroup S is said to be completely prime if for each x,y∈S and γ∈Γ such that xγy∩K≠∅, then x∈K or y∈K. An ordered Γ-semihypergroup S,Γ,≤ is called m,n-regular if for every a∈S there exist x∈S and α,β∈Γ such that a≤amαxβan.Theorem 5.
LetF be a nonempty subset of a commutative m,n-regular ordered Γ-semihypergroup S,Γ,≤. Then, the following assertions are equivalent:(1)
F is an m,n−Γ-hyperfilter of S.(2)
SF=∅ or S\F is a completely prime generalized m,n−Γ-hyperideal of S.Proof.
(1)
⇒2: Assume that (1) holds. Let S\F≠∅. First of all, we show that S\F is a generalized m,n−Γ-hyperideal of S, i.e., S\FmΓSΓS\Fn⊆S\F. Let S\FmΓSΓS\Fn⊆F. Since S is m,n-regular and F=F, we get(26)S\F⊆S\FmΓSΓS\Fn⊆F=F,which is a contradiction. Suppose that a∈S\F and x≤a, where x∈S. If x∈F, then, since F is an m,n−Γ-hyperfilter of S, we get a∈F, a contradiction. It implies that x∈S\F, and hence S\F⊆S\F. Thus, S\F is a generalized m,n−Γ-hyperideal of S. Now, we assert that S\F is a completely prime generalized m,n−Γ-hyperideal of S. Let x,y∈S, γ∈Γ, and xγy∩S\F≠∅. Then, there exists a∈xγy such that a∈S\F. If x∈F and y∈F, then, since F is a sub Γ-semihypergroup of S, we get xγy⊆F, a contradiction. Hence, x∈S\F or y∈S\F. Therefore, S\F is a completely prime generalized m,n−Γ-hyperideal of S.(2)
⇒1: If S\F=∅, then F=S, and hence F is an m,n−Γ-hyperfilter of S. Let S\F be a completely prime generalized m,n−Γ-hyperideal of S. Now, let x,y∈F and γ∈Γ. If xγy⊈F, then xγy∩S\F≠∅. Since S\F is completely prime, we get x∈S\F or y∈S\F, which is a contradiction. Thus, xγy⊆F and so F is a sub Γ-semihypergroup of S. Now, let a,b∈S, γ∈Γ, and aγb∩F≠∅. Let am⊆S\F or bn⊆S\F. Without any loss of generality, we assume that am⊆S\F. Since S is m,n-regular, we get a≤amαxβan. Then,(27)aγb≤amαxβanγb≤ammλyμamnαxβanγb.
SinceS is commutative and S\F is an m,n−Γ-hyperideal of S, we have(28)aγb⊆S\FmΓSΓS\Fn⊆S\F.
It implies thatS\F∩F≠∅, which is a contradiction. Thus, am⊆F and bn⊆F. Next, let a∈F and a≤c∈S. If c∈S\F, then, since S\F is an m,n−Γ-hyperideal of S, we get a∈S\F, a contradiction. So, c∈F, and thus F is an m,n−Γ-hyperfilter of S.


## 4. Construction of OrderedΓ-Semihypergroups

In this section, we consider some orderedΓ-semihypergroups, where we define an ordered regular equivalence relation σ∗ such that the quotient is an ordered Γ-semihypergroup. More exactly, starting with an ordered Γ-semihypergroup and using σ∗, we can construct an ordered Γ-semihypergroup structure on the quotient set.Example 3.
Let(29)S=w,x,y,z.
Define the hyperoperationΓ=γ and (partial) order relation ≤ on S as follows:(30)
Then,S,Γ,≤ is an ordered Γ-semihypergroup. We give the covering relation ≺ as(31)≺=w,x,w,y,x,z,y,z.
The figure ofS is shown in Figure 3.
Here, it is a routine matter to verify thatK=w,y is an A−I−Γ-hyperideal of S.Figure 3
Figure ofS for Example 3.Example 4.
Suppose that(32)S=a,b,c,d,e,and Γ=β. Define(33)
We consider the orderedΓ-semihypergroup S,Γ,≤, where the (partial) order relation ≤ is defined by the following table:(34)≤≔a,a,a,b,b,b,c,c,d,b,d,c,d,d,e,e.
We give the covering relation≺ as(35)≺=a,b,d,b,d,c.
The figure ofS is shown in Figure 4.
Suppose thatF=a,b,c,d. Clearly, F is an m,n−Γ-hyperfilter of S for m,n≥2. Indeed: since eβx=c or d∈F, xβe=c or d∈F for any x∈S and em=c⊆F for m≥2, we have that F is an m,n−Γ-hyperfilter of S for m,n≥2.
ButF is not a Γ-hyperfilter of S. Indeed: since eβx=c or d∈F, xβe=c or d∈F for any x∈S and e∉ℱ, we have that F is not a Γ-hyperfilter of S.Figure 4
Figure ofS for Example 4.Definition 6.
LetS,Γ,≤ be an ordered Γ-semihypergroup. A relation σ on S is called a weak pseudoorder on S if(1)
≤⊆σ.(2)
aσb and bσc imply aσc for all a,b,c∈S.(3)
aσb implies aγcσ⟶bγc and cγaσ⟶cγb for all a,b,c∈S and γ∈Γ.(4)
aσb and bσa imply aγcσ˜bγc and cγaσ˜cγb for all a,b,c∈S and γ∈Γ.
Clearly, every pseudoorder relation on an orderedΓ-semihypergroup S is a weak pseudoorder. The converse is not true, in general, that is, a weak pseudoorder may not be a pseudoorder of S (see Example 6).Theorem 6.
LetS,Γ,≤ be an ordered Γ-semihypergroup and σ a weak pseudoorder on S. Then, there exists a regular equivalence relation(36)σ∗=a,b∈S×S|aσbandbσa,on S such that S/σ∗,Γσ∗,≺¯σ∗ is an ordered Γσ∗-semihypergroup, where(37)≺¯σ∗≔σ∗x,σ∗y∈Sσ∗×Sσ∗|∃a∈σ∗x,∃b∈σ∗y,such that a,b∈σ.Proof.
The proof is similar to the proof of Theorem4 in [16].Example 5.
Let(38)S=a,b,c,d,e,and Γ=γ,β. Define the hyperoperations γ,β and (partial) order relation ≤ on S as follows:(39)
Then,S,Γ,≤ is an ordered Γ-semihypergroup. We give the covering relation ≺ as(40)≺=a,b,b,c.
The figure ofS is shown in Figure 5.
Here,F=a,b,c is a Γ-hyperfilter of S. Let us consider the following relation in S:(41)θ=a,a,a,b,a,c,b,b,b,c,c,b,c,c,d,d,d,e,e,d,e,e.
Now, we consider a regular equivalence relationθ∗ on S as follows:(42)θ∗=x,y∈S×S|x,y∈θandy,x∈θ.
By definition ofθ∗, we get(43)θ∗=a,a,b,b,b,c,c,b,c,c,d,d,d,e,e,d,e,e.
If we take(44)w1=a,w2=b,c,w3=d,e,then, by Theorem 6, S/θ∗=w1,w2,w3 is an ordered Γθ∗-semihypergroup, where Γθ∗=γθ∗,βθ∗ and ≺¯θ∗ are defined by(45)such that a,b∈θ=w1,w1,w1,w2,w2,w2,w3,w3.
We give the covering relation≺θ∗ as(46)≺θ∗=w1,w2.
The figure ofS/θ∗ is shown in Figure 6.Figure 5
Figure ofS for Example 5.Figure 6
Figure ofS/θ∗ for Example 5.Example 6.
Let(47)S=a,b,c,d,e,f,and Γ=γ,β be the sets of binary hyperoperations defined as follows:(48)
We have thatS,Γ,≤ is an ordered Γ-semihypergroup where the (partial) order relation ≤ is defined by(49)≤≔a,a,b,a,b,b,b,c,c,c,d,d,e,e,f,f.
We give the covering relation≺ as(50)≺=b,a,b,c.
The figure ofS is shown in Figure 7.
S has no proper m,n−Γ-hyperfilter. Now, we put(51)ρ≔a,a,a,b,a,c,b,a,b,b,b,c,c,c,d,d,e,e,f,f.
Clearly,ρ is not a pseudoorder, since aγfρ¯¯cγf does not hold. Indeed:(52)a,c∈ρ,aγf=a,cγf=a,fbuta,f∉ρ.
We have(53)ρ∗=x,y∈S×S|x,y∈ρ,y,x∈ρ=a,a,a,b,b,a,b,b,c,c,d,d,e,e,f,f.
Then,S/ρ∗=u1,u2,u3,u4,u5, where u1=a,b, u2=c, u3=d, u4=e, and u5=f. By Theorem 6, S/ρ∗,Γρ∗,≺¯ρ∗ is an ordered Γρ∗-semihypergroup, where γρ∗, βρ∗, and ≺¯ρ∗ are defined by(54)
We give the covering relation≺ρ∗ as(55)≺ρ∗=u1,u2.
The figure ofS/ρ∗ is shown in Figure 8.
In the following example, we show that the converse of Lemma1 is not true in general, i.e., an A−I−Γ-hyperideal may not be an I−Γ-hyperideal of an ordered Γ-semihypergroup S.Figure 7
Figure ofS for Example 6.Figure 8
Figure ofS/ρ∗ for Example 6.Example 7.
We come back to Example6 and consider ordered Γ-semihypergroup S,Γ,≤. Put K=a,b,c. Clearly, K is a sub Γ-semihypergroup of S. We havexΓKΓy∩K≠∅ for all x,y∈S.SΓKΓS=S⊈K.K=K.
Therefore,K is an A−I−Γ-hyperideal of S but is not I−Γ-hyperideal of S.


## 5. Conclusions

One of the most important research areas in orderedΓ-semihypergroup theory is the investigation of hyperfilters [25]. Generalization of hyperfilters in (ordered) Γ-semihypergroups is necessary for further study of (ordered) Γ-semihypergroups. In this paper, we studied some properties of A−I−Γ-hyperideals and m,n−Γ-hyperfilters of ordered Γ-semihypergroups. For future work, it will be interesting to study relative A−I−Γ-hyperideals and intuitionistic fuzzy A−I−Γ-hyperideals in ordered hyperstructures such as ordered Γ-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 m,n−Γ-hyperfilters and ∈,∈∨qk-fuzzy m,n−Γ-hyperfilters.



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