# Rough Hyperfilters in Po-LA-Semihypergroups

Authors: Ferdaous Bouaziz; Naveed Yaqoob
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper concerns the study of hyperfilters of ordered LA-semihypergroups, and presents some examples in this respect. Furthermore, we study the combination of rough set theory and hyperfilters of an ordered LA-semihypergroup. We define the concept of rough hyperfilters and provide useful examples on it. A rough hyperfilter is a novel extension of hyperfilter of an ordered LA-semihypergroup. We prove that the lower approximation of a left (resp., right, bi) hyperfilter of an ordered LA-semihypergroup becomes left (resp., right, bi) hyperfilter of an ordered LA-semihypergroup. Similarly we prove it for upper approximation.


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## Body

## 1. Introduction

Theory of hyperstructures (also known as multialgebras) was introduced in 1934 at the “8th congress of Scandinavian Mathematicians”, where a French mathematician Marty [1] presented some definitions and results on the hypergroup theory, which is a generalization of groups. He gave applications of hyperstructures to rational functions, algebraic functions and noncommutative groups. Hyperstructure theory is an extension of the classical algebraic structure. In algebraic hyperstructures, the composition of two elements is a set, while in a classical algebraic structure, the composition of two elements is an element. Algebraic hyperstructure theory has many applications in other disciplines. Because of new viewpoints and importance of this theory, many authors applied this theory in nuclear physics and chemistry. Around 1940s, several mathematicians added many results to the theory of hyperstructures, especially in United States, France, Russia, Italy, Japan, and Iran. Several papers and books have been written on hypergroups. One of these is “Applications of hyperstructure theory” written by Corsini and Leoreanu-Fotea [2]. Another book “Hyperstructures and Their Representations” written by Vougiouklis [3], was published in 1994. Recently, Hila et al. [4] and Yaqoob et al. [5] introduced the concept of nonassociative semihypergroups which is a generalization of the work of Kazim and Naseeruddin [6]. Later, Yaqoob and Gulistan [7] studied partially ordered left almost semihypergroups.Filter theory in algebraic structures has been studied by many mathematicians. Hila [8] studied filters in ordered Γ-semigroups. Jakubík [9] studied filters of ordered semigroups. Jirojkul and Chinram [10] studied fuzzy quasi-ideal subsets and fuzzy quasi-filters of ordered semigroup. Jun et al. [11] studied bifilters of ordered semigroups. Kehayopulu [12] introduced filters generated in poe-semigroups. Ren et al. [13] introduced principal filters of po-semigroups. Tang et al. [14] studied hyperfilters of ordered semihypergroups.Rough set theory, introduced in 1982 by Pawlak [15], is a mathematical approach to imperfect knowledge. The methodology of rough set is concerned with the classification and analysis of imprecise, uncertain, or incomplete information and knowledge. There are several authors who considered the theory of rough sets in algebraic structures, for instance, Biswas and Nanda [16] developed some results on rough subgroups. Jun [17] applied the rough set theory to gamma-subsemigroups/ideals in gamma-semigroups. Shabir and Irshad [18] defined roughness in ordered semigroups. Rough set theory is also considered by some authors in different hyperstructures, for instance, in semihypergroups [19], Γ-semihypergroups [20], hyperrings [21], Hv-groups [22], Hv-modules [23], Γ-semihyperrings [24], hyperlattices [25], hypergroup [26, 27], quantales [28], quantale modules [29], and nonassociative po-semihypergroups [30]. The rough set theory has been applied to the filters of some well known algebraic structures, like BE-algebras [31], ordered semigroups [32, 33], residuated lattices [34], and BL-algebras [35].


## 2. Preliminaries and Basic Definitions

Definition 1 [4, 5].
A hypergroupoidH,∘ is said to be an LA-semihypergroup if for all a,b,c∈H,(1)a∘b∘c=c∘b∘a.Definition 2 [7].
LetH be a nonempty set and “≤” be an ordered relation on H. Then H,∘,≤ is called an ordered LA-semihypergroup if(1)
H,∘ is an LA-semihypergroup;(2)
H,≤ is a partially ordered set;(3)
for everya,b,c∈H, a≤b implies a∘c≤b∘c and c∘a≤c∘b, where a∘c≤b∘c means that for every x∈a∘c there exists y∈b∘c such that x≤y.Definition 3 [7].
IfA is a nonempty subset of H,∘,≤, then A is the subset of H defined as follows:(2)A=t∈H:t≤a,forsomea∈A.Definition 4 [7].
A nonempty subsetA of an ordered LA-semihypergroup H,∘,≤ is called an LA-subsemihypergroup of H if A∘A⊆A.Definition 5 [7].
A nonempty subsetA of an ordered LA-semihypergroup H,∘,≤ is called a right (resp., left) hyperideal of H if(1)
A∘H⊆Aresp.,H∘A⊆A,(2)
for everya∈H, b∈A and a≤b implies a∈A.
IfA is both right hyperideal and left hyperideal of H, then A is called a hyperideal (or two sided hyperideal) of H.Definition 6 [7].
A nonempty subsetP of H,∘,≤ is called a prime hyperideal of H if the following conditions hold:(i)
A∘B⊆P⇒A⊆P or B⊆P for any two hyperideals A and B of H;(ii)
ifa∈P and b≤a, then b∈P for every b∈H.Definition 7 [30].
A relationδ on an ordered LA-semihypergroup H is called a pseudohyperorder if(1)
≤⊆δ.(2)
δ is transitive, that is a,b,b,c∈δ implies a,c∈δ for all a,b,c∈H.(3)
δ is compatible, that is if a,b∈δ then a∘x,b∘x∈δ and x∘a,x∘b∈δ for all a,b,x∈H.


## 3. Hyperfilters in Ordered LA-Semihypergroups

Definition 8.
LetH,∘,≤ be an ordered LA-semihypergroup. An LA-subsemihypergroup F of H is called a left (resp., right) hyperfilter of H if(i)
for alla,b∈H, a∘b⊆F⇒a∈F (resp., b∈F),(ii)
for allb∈H and a∈F, a≤b⇒b∈F.Definition 9.
LetH,∘,≤ be an ordered LA-semihypergroup. An LA-subsemihypergroup F of H is called a hyperfilter of H if(i)
for alla,b∈H, a∘b⊆F⇒a∈F and b∈F,(ii)
for allb∈H and a∈F, a≤b⇒b∈F.Example 10.
Consider a setH=a,b,c with the following hyperoperation “∘” and the order “≤”:
We give the covering relation “≺” as:(3)≺≔c,a.
The figure ofH is shown in Figure 1. Then H,∘,≤ is an ordered LA-semihypergroup. Here a,b,a,c and H are hyperfilters of H.Figure 1
Figure ofH for Example 10.Definition 11.
LetH,∘,≤ be an ordered LA-semihypergroup. An LA-subsemihypergroup F of H is called a bi-hyperfilter of H if(i)
for alla,b∈H, a∘b∘a⊆F⇒a∈F,(ii)
for allb∈H and a∈F, a≤b⇒b∈F.Example 12.
Consider a setH=a,b,c,d,e with the following hyperoperation “∘” and the order “≤”:
We give the covering relation “≺” as:(4)≺=a,b,a,c,a,d.
The figure ofH is shown in Figure 2. Then H,∘,≤ is an ordered LA-semihypergroup. The bi-hyperfilters of H are e, b,c,e and H.Figure 2
Figure ofH for Example 12.Definition 13.
LetH,∘,≤ be an ordered LA-semihypergroup. An LA-subsemihypergroup F of H is called a strong left (resp., strong right) hyperfilter of H if(i)
for alla,b∈H, a∘b∩F≠∅⇒a∈F (resp., b∈F),(ii)
for allb∈H and a∈F, a≤b⇒b∈F.Definition 14.
LetH,∘,≤ be an ordered LA-semihypergroup. An LA-subsemihypergroup F of H is called a strong hyperfilter of H if(i)
for alla,b∈H, a∘b∩F≠∅⇒a∈F and b∈F,(ii)
for allb∈H and a∈F, a≤b⇒b∈F.Example 15.
Consider a setH=a,b,c,d,e with the following hyperoperation “∘” and the order “≤”:
We give the covering relation “≺” as:(5)≺=c,b,d,b,e,b.
The figure ofH is shown in Figure 3. Then H,∘,≤ is an ordered LA-semihypergroup. The hyperfilters of H are a, a,b and H. Here a and H are also strong hyperfilters of H. While a,b is not a strong hyperfilter of H, because d∘e∩a,b=b≠∅, but d∉a,b and e∉a,b.Figure 3
Figure ofH for Example 15.Definition 16.
LetH,∘,≤ be an ordered LA-semihypergroup. An LA-subsemihypergroup F of H is called a strong bi-hyperfilter of H if(i)
for alla,b∈H, a∘b∘a∩F≠∅⇒a∈F,(ii)
for allb∈H and a∈F, a≤b⇒b∈F.Example 17.
Consider a setH=a,b,c with the following hyperoperation “∘” and the order “≤”:
We give the covering relation “≺” as:(6)≺=a,b,a,c.
The figure ofH is shown in Figure 4. Then H,∘,≤ is an ordered LA-semihypergroup. The bi-hyperfilters of H are a, b,c and H. But H is the only strong bi-hyperfilter of H.Figure 4
Figure ofH for Example 17.Lemma 18.
LetH,∘,≤ be an ordered LA-semihypergroup and S be an LA-subsemihypergroup of H. Then for a left (resp., right, bi) hyperfilter F of H, either F∩S=∅ or F∩S is a left (resp., right, bi) hyperfilter of S.Proof.
ConsiderS is an LA-subsemihypergroup of H and F is a bi-hyperfilter of H. Suppose that F∩S=∅. Now F∩S2⊆F2⊆F and F∩S2⊆S2⊆S. Therefore, F∩S2⊆F∩S. So F∩S is an LA-subsemihypergroup of S. Suppose that, for any a,b∈S, a∘b∘a⊆F∩S. Therefore, a∘b∘a⊆F. As a∈S and F is a bi-hyperfilter of H, then a∈F. Thus, a∈F∩S. Finally take any a∈F∩S and b∈S such that a≤b. As F is a bi-hyperfilter of H with a∈F and a≤b∈S, b∈F. Therefore, b∈F∩S. Hence F∩S is a bi-hyperfilter of S. The other cases can be seen in similar way.Lemma 19.
LetH,∘,≤ be an ordered LA-semihypergroup and S be an LA-subsemihypergroup of H. Then for a strong left (resp., strong right, strong bi) hyperfilter F of H, either F∩S=∅ or F∩S is a strong left (resp., strong right, strong bi) hyperfilter of S.Proof.
Proof is straightforward.Theorem 20.
LetH,∘,≤ be an ordered LA-semihypergroup and Fi|i∈I a family of left (resp., right, bi) hyperfilters of H. Then ⋂i∈IFi is a left (resp., right, bi) hyperfilter of H if ⋂i∈IFi≠∅, where |I|≥2.Proof.
ConsiderFi is a bi-hyperfilter of H for each i∈I. Assume that ⋂i∈IFi≠∅. Let a,b∈⋂i∈IFi. Then a,b∈Fi for all i∈I. As, for all i∈I, Fi is a bi-hyperfilter of H, so a∘b⊆Fi. Hence a∘b⊆⋂i∈IFi, i.e., ⋂i∈IFi is an LA-subsemihypergroup of H. Suppose that, for any a,b∈H, a∘b∘a⊆⋂i∈IFi. Therefore, a∘b∘a⊆Fi for all i∈I. As Fi is a bi-hyperfilter of H, then a∈Fi for all i∈I. Thus, a∈⋂i∈IFi. Finally take any a∈⋂i∈IFi and b∈H such that a≤b. Therefore, a∈Fi for all i∈I. As Fi is a bi-hyperfilter of H with a∈Fi and a≤b∈S, this implies b∈Fi for all i∈I. Therefore, b∈⋂i∈IFi. Hence ⋂i∈IFi is a bi-hyperfilter of H. The other cases can be seen in similar way.Theorem 21.
LetH,∘,≤ be an ordered LA-semihypergroup and Fi|i∈I a family of strong left (resp., strong right, strong bi) hyperfilters of H. Then ⋂i∈IFi is a strong left (resp., strong right, strong bi) hyperfilter of H if ⋂i∈IFi≠∅, where |I|≥2.Proof.
Proof is straightforward.Remark 22.
Generally, the union of two hyperfilters of an ordered LA-semihypergroupH need not be a hyperfilter of H.The following example shows that, the union of two hyperfilters of an ordered LA-semihypergroupH is not a hyperfilter of H.Example 23.
Consider Example1. One can easily see that F1=a and F2=b are both hyperfilters of H. But F1∪F2=a,b is not a hyperfilter of H, because F1∪F22⊈F1∪F2, i.e., F1∪F2 is not an LA-subsemihypergroup of H.Lemma 24.
LetH,∘,≤ be an ordered LA-semihypergroup and F1, F2 hyperfilters of H. Then F1∪F2 is a hyperfilter of H if and only if F1⊆F2 or F2⊆F1.Proof.
Proof is straightforward.Theorem 25.
LetH be an ordered LA-semihypergroup and F be a nonempty subset of H. Then the following statements are equivalent;(i)
F is a left hyperfilter of H.(ii)
H\F=∅ or H\F is a prime right hyperideal of H.Proof.
(i)⇒ (ii) Assume that H\F≠∅. Let x∈H\F and y∈H. If x∘y⊈H\F, then x∘y⊆F. Since F is a left hyperfilter, so x∈F which is not possible. Thus, x∘y⊆H\F, and so H\F∘H⊂H\F. In order to show that H\F is right hyperideal of H, we need to show that H\F∘H⊆H\F. Let x∈H\F∘H, which implies that x∈y∘z for some y∈H\F and z∈H. Now consider x≤y∘z⊆H\F. Thus, H\F∘H⊆H\F, which shows that H\F is right hyperideal of H.
Next we shall show thatH\F is prime. Let x∘y⊆H\F for x,y∈H. Suppose that x∉H\F and y∉H\F. Then x∈F and y∈F. Since F is an LA-subsemihypergroup of H, so x∘y⊆F. It is not possible. Thus, x∈H\F and y∈H\F. Hence, H\F is prime, and so H\F is a prime right hyperideal.
(ii)⇒ (i) If H\F=∅, then F=H. Thus, in this case, F is a left hyperfilter of H. Next assume that H\F is a prime right hyperideal, then F is an LA-subsemihypergroup of H. Suppose that x∘y⊈F for x,y∈F. Then x∘y⊆H\F for x,y∈F. Since H\F is prime, x∈H\F and y∈H\F. It is not possible. Thus, x∘y⊆F and so F is an LA-subsemihypergroup of H. Let x∘y⊆F for x,y∈H. Then, x∈F. Indeed, if x∉F, then x∈H\F. Since H\F is prime right hyperideal of H, x∘y⊆H\F∘H⊂H\F. It is not possible. Thus, x∈F. Therefore, F is a left hyperfilter of H.Theorem 26.
LetH be an ordered LA-semihypergroup and F be a nonempty subset of H. Then the following statements are equivalent;(1)
F is a right hyperfilter of H.(2)
H\F=∅ or H\F is a prime left hyperideal.Proof.
The proof is similar to the proof of Theorem25.Theorem 27.
LetH be an ordered LA-semihypergroup and F be a nonempty subset of H. Then the following statements are equivalent;(1)
F is a hyperfilter of H.(2)
H\F=∅ or H\F is a prime hyperideal of H.Proof.
Follows directly from Theorems25 and 26.


## 4. Rough Hyperfilters in Ordered LA-Semihypergroups

In this section, we will study some results on hyperfilters of ordered LA-semihypergroups in terms of rough sets which are based on pseudohyperorder relation.Definition 28 [30].
LetX be a nonempty set and δ be a binary relation on X. By ℘X we mean the power set of X. For all A⊆X, we define δ− and δ+:℘X→℘X by(7)δ−A=x∈X:∀y,xδy⇒y∈A=x∈X:δNx⊆A, and (8)δ+A=x∈X:∃y∈A,suchthatxδy=x∈X:δNx∩A≠∅,
whereδNx=y∈X:xδy. δ−A and δ+A are called the lower approximation and the upper approximation operations, respectively.Definition 29 [30, Definition 4.1].
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset A of H is called a δ-upper (resp., δ-lower) rough LA-subsemihypergroup of H if δ+A (resp., δ−A) is an LA-subsemihypergroup of H.Theorem 30 [30, Theorem 4.3].
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H and A an LA-subsemihypergroup of H. Then(1)
δ+A is an LA-subsemihypergroup of H.(2)
ifδ is complete, then δ−A is, if it is nonempty, an LA-subsemihypergroup of H.Definition 31.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-lower rough left hyperfilter (resp., right hyperfilter, hyperfilter) of H if δ−F is a left hyperfilter (resp., right hyperfilter, hyperfilter) of H.Theorem 32.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a left hyperfilter (resp., right hyperfilter, hyperfilter) of H. Then δ−F is, if it is nonempty, a left hyperfilter (resp., right hyperfilter, hyperfilter) of H.Proof.
LetF be a left hyperfilter of H. Since F is an LA-subsemihypergroup of H, then by Theorem 30(2), δ−F is an LA-subsemihypergroup of H. Let x∘y⊆δ−F for some x,y∈H. Then δNx∘δNy=δNx∘y⊆F. We suppose that δ−F is not a left hyperfilter of H. Then there exist x,y∈H such that x∘y⊆δ−F but x∉δ−F. Thus, δNx⊈F. Then there exist a∈δNx such that a∉F and b∈δNy. Thus, a∘b⊆δNx∘δNy⊆F. Since F is a left hyperfilter of H, so we have a∈F, which is a contradiction. Now, let x∈δ−F and y∈H such that x≤y. Then δNx⊆F and xδy. This implies that δNx=δNy. Since δNx⊆F, so δNy⊆F. Thus, y∈δ−F. Thus, δ−F is a left hyperfilter of H. The case for right hyperfilter and hyperfilter can be proved in a similar way.Definition 33.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-upper rough left hyperfilter (resp., right hyperfilter, hyperfilter) of H if δ+F is a left hyperfilter (resp., right hyperfilter, hyperfilter) of H.Theorem 34.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a left hyperfilter (resp., right hyperfilter, hyperfilter) of H. Then δ+F is a left hyperfilter (resp., right hyperfilter, hyperfilter) of H.Proof.
LetF be a left hyperfilter of H. Since F is an LA-subsemihypergroup of H, then by Theorem 30(1), δ+F is an LA-subsemihypergroup of H. Let x∘y⊆δ+F for some x,y∈H. Then δNx∘δNy∩F=δNx∘y∩F≠∅, so there exist a∈δNx and b∈δNy such that a∘b⊆F. Since F is left hyperfilter of H, so we have a∈F. Thus, δNx∩F≠∅ which implies that x∈δ+F. Now, let x∈δ+F and y∈H such that x≤y. Then δNx∩F≠∅ and xδy. This implies that δNx=δNy. Since δNx∩F≠∅, so δNy∩F≠∅. Thus, y∈δ+F. Thus, δ+F is a left hyperfilter of H, that is, F is a δ-upper rough left hyperfilter of H. The case for right hyperfilter and hyperfilter can be proved in a similar way.The converse of Theorems32 and 34 is not true in general.Example 35.
Consider a setH=a,b,c,d,e with the following hyperoperation “∘” and the order “≤”:
We give the covering relation “≺” as:(9)≺=d,e.
The figure ofH is shown in Figure 5. Then H,∘,≤ is an ordered LA-semihypergroup. Now let(10)δ=a,a,b,b,b,c,c,b,c,c,d,d,d,e,e,d,e,e
be a complete pseudohyperorder onH, such that(11)δNa=a,δNb=δNc=b,candδNd=δNe=d,e.
Now fora,b,d⊆H,(12)δ−a,b,d=aandδ+a,b,d=a,b,c,d,e.
It is clear thatδ−a,b,d and δ+a,b,d are both hyperfilters of H, but a,b,d is not a hyperfilter of H.Figure 5
Figure ofH for Example 35.Definition 35.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-lower rough bi-hyperfilter of H if δ−F is a bi-hyperfilter of H.Theorem 36.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a bi-hyperfilter of H. Then δ−F is, if it is nonempty, a bi-hyperfilter of H.Proof.
LetF be a bi-hyperfilter of H. Since F is an LA-subsemihypergroup of H, then by Theorem 30(2), δ−F is an LA-subsemihypergroup of H. Let x∘y∘x⊆δ−F for some x,y∈H. Then δNx∘δNy∘δNx=δNx∘y∘x⊆F. We suppose that δ−F is not a bi-hyperfilter of H. Then there exist x,y∈H such that x∘y∘x⊆δ−F, but x∉δ−F. Thus, δNx⊈F. Then there exist a∈δNx such that a∉F and b∈δNy. Thus, a∘b∘a⊆δNx∘δNy∘δNx⊆F. Since F is a bi-hyperfilter of H, so we have a∈F, which is a contradiction. Now, let x∈δ−F and y∈H such that x≤y. Then δNx⊆F and xδy. This implies that δNx=δNy. Since δNx⊆F, so δNy⊆F. Thus y∈δ−F. Hence δ−F is a bi-hyperfilter of H.Definition 37.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-upper rough bi-hyperfilter of H if δ+F is a bi-hyperfilter of H.Theorem 38.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a bi-hyperfilter of H. Then δ+F is a bi-hyperfilter of H.Proof.
LetF be a bi-hyperfilter of H. Since F is an LA-subsemihypergroup of H, then by Theorem 30(1), δ+F is an LA-subsemihypergroup of H. Let x∘y∘x⊆δ+F for some x,y∈H. Then δNx∘δNy∘δNx∩F=δNx∘y∘x∩F≠∅, so there exist a∈δNx and b∈δNy such that a∘b∘a⊆F. Since F is bi-hyperfilter of H, so we have a∈F. Thus, δNx∩F≠∅ which implies that x∈δ+F. Now, let x∈δ+F and y∈H such that x≤y. Then δNx∩F≠∅ and xδy. This implies that δNx=δNy. Since δNx∩F≠∅, so δNy∩F≠∅. Thus, y∈δ+F. Thus, δ+F is a bi-hyperfilter of H, that is, F is a δ-upper rough bi-hyperfilter of H.Definition 39.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-lower rough strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H if δ−F is a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H.Theorem 40.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H. Then δ−F is, if it is nonempty, a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H.Proof.
LetF be a strong hyperfilter of H. Since F is an LA-subsemihypergroup of H, then by Theorem 30(2), δ−F is an LA-subsemihypergroup of H. Let x∘y∩δ−F≠∅ for some x,y∈H. For t∈x∘y∩δ−F, we have t∈x∘y and t∈δ−F. This implies that δNt⊆δNx∘y=δNx∘δNy and δNt⊆F. This implies that δNt⊆δNx∘δNy∩F. Thus δNx∘δNy∩F≠∅. We suppose that δ−F is not a strong hyperfilter of H. Then there exist x,y∈H such that x∘y∩δ−F≠∅ but x∉δ−F and y∉δ−F. Thus, δNx⊈F and δNy⊈F. Then there exist a∈δNx and b∈δNy such that a∉F and b∉F. Thus, a∘b∩F⊆δNx∘δNy∩F≠∅. Since F is a strong hyperfilter of H, so we have a∈F and b∈F, which is a contradiction. Now, let x∈δ−F and y∈H such that x≤y. Then δNx⊆F and xδy. This implies that δNx=δNy. Since δNx⊆F, so δNy⊆F. Thus, y∈δ−F. Hence δ−F is a strong hyperfilter of H. The case for strong left hyperfilter and strong right hyperfilter can be proved in a similar way.Definition 41.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-upper rough strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H if δ+F is a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H.Theorem 42.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H. Then δ+F is a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of H.Proof.
LetF be a strong hyperfilter of H. Since F is an LA-subsemihypergroup of H, then by Theorem 30(1), δ+F is an LA-subsemihypergroup of H. Let x∘y∩δ+F≠∅ for some x,y∈H. Then δNx∘δNy∩F=δNx∘y∩F≠∅, so there exist a∈δNx and b∈δNy such that a∘b∩F≠∅. Since F is strong hyperfilter of H, so we have a∈F and b∈F. Thus, δNx∩F≠∅ and δNy∩F≠∅, which implies that x∈δ+F and y∈δ+F. Now, let x∈δ+F and y∈H such that x≤y. Then δNx∩F≠∅ and xδy. This implies that δNx=δNy. Since δNx∩F≠∅, so δNy∩F≠∅. Thus, y∈δ+F. Thus, δ+F is a strong hyperfilter of H, that is, F is a δ-upper rough strong hyperfilter of H. The case for strong left hyperfilter and strong right hyperfilter can be proved in a similar way.The converse of Theorems39 and 41 is not true in general.Example 43.
Consider a setH=a,b,c,d,e with the following hyperoperation “∘” and the order “≤”:
We give the covering relation “≺” as:(13)≺=a,b,e,c,e,d.
The figure ofH is shown in Figure 6. Then H,∘,≤ is an ordered LA-semihypergroup. Now let δ be a complete pseudohyperorder on H, such that(14)δNa=δNb=a,bandδNc=δNd=δNe=c,d,e.
Now fora,b,c⊆H,(15)δ−a,b,c=a,bandδ+a,b,c=a,b,c,d,e.
It is clear thatδ−a,b,c and δ+a,b,c are both strong hyperfilters of H but a,b,c is not a hyperfilter of H.Definition 44.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-lower rough strong bi-hyperfilter of H if δ−F is a strong bi-hyperfilter of H.Theorem 45.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a strong bi-hyperfilter of H. Then δ−F is, if it is nonempty, a strong bi-hyperfilter of H.Figure 6
Figure ofH for Example 42.Proof.
The proof is similar to the proof of Theorem40.Definition 46.
Letδ be a pseudohyperorder on an ordered LA-semihypergroup H. Then a nonempty subset F of H is called a δ-upper rough strong bi-hyperfilter of H if δ+F is a strong bi-hyperfilter of H.Theorem 47.
Letδ be a complete pseudohyperorder on an ordered LA-semihypergroup H and F a strong bi-hyperfilter of H. Then δ+F is a strong bi-hyperfilter of H.Proof.
The proof is similar to the proof of Theorem42.



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