Research ArticleRough Hyperfilters in Po-LA-Semihypergroups

Ferdaous Bouaziz1 and Naveed Yaqoob 2

1Department of Mathematics, College of Science and Arts Buraydha, Qassim University, Saudi Arabia2Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad, Pakistan

Correspondence should be addressed to Naveed Yaqoob; nayaqoob@ymail.com

Received 4 July 2019; Revised 30 August 2019; Accepted 13 September 2019; Published 29 December 2019

Academic Editor: Xiaohua Ding

Copyright © 2019 Ferdaous Bouaziz and Naveed Yaqoob. �is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

�eory 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 de�nitions and results on the hypergroup the-ory, 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 ele-ments 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 the-ory in nuclear physics and chemistry. Around 1940s, several mathematicians added many results to the theory of hyper-structures, 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 �eir 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 le� almost semihypergroups.

Filter theory in algebraic structures has been studied by many mathematicians. Hila [8] studied �lters in ordered Γ-semigroups. Jakubík [9] studied �lters of ordered semi-groups. Jirojkul and Chinram [10] studied fuzzy quasi-ideal subsets and fuzzy quasi-�lters of ordered semigroup. Jun et al. [11] studied bi�lters of ordered semigroups. Kehayopulu [12] introduced �lters generated in poe-semigroups. Ren et al. [13] introduced principal �lters of po-semigroups. Tang et al. [14] studied hyper�lters of ordered semihypergroups.

Rough set theory, introduced in 1982 by Pawlak [15], is a mathematical approach to imperfect knowledge. �e methodol-ogy of rough set is concerned with the classi�cation and analysis of imprecise, uncertain, or incomplete information and knowl-edge. �ere 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 gam-ma-semigroups. Shabir and Irshad [18] de�ned roughness in ordered semigroups. Rough set theory is also considered by some authors in di¯erent hyperstructures, for instance, in semihyper-groups [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]. �e rough set theory has been applied to the �lters of some well known algebraic struc-tures, like BE-algebras [31], ordered semigroups [32, 33], resid-uated lattices [34], and BL-algebras [35].

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019, Article ID 8326124, 8 pageshttps://doi.org/10.1155/2019/8326124

Discrete Dynamics in Nature and Society2

2. Preliminaries and Basic Definitions

De­nition 1 [4, 5]. A hypergroupoid (�, ∘) is said to be an LA-semihypergroup if for all �, �, � ∈ �,

De­nition 2 [7]. Let � be a nonempty set and “≤” be an ordered relation on �. �en (�, ∘, ≤) is called an ordered LA-semihypergroup if

(1) (�, ∘) is an LA-semihypergroup;(2) (�, ≤) is a partially ordered set;(3) for every �, �, � ∈ �, � ≤ � implies � ∘ � ≤ � ∘ � and � ∘ � ≤ � ∘ �, where � ∘ � ≤ � ∘ � means that for every � ∈ � ∘ � there exists � ∈ � ∘ � such that � ≤ �.

De­nition 3 [7]. If � is a nonempty subset of (�, ∘, ≤), then (�] is the subset of � de�ned as follows:

De­nition 4 [7]. A nonempty subset � of an ordered LA-semihypergroup (�, ∘, ≤) is called an LA-subsemihypergroup of � if (� ∘ �] ⊆ (�].

De­nition 5 [7]. A nonempty subset � of an ordered LA-semihypergroup (�, ∘, ≤) is called a right (resp., le�) hyperideal of � if

(1) � ∘ � ⊆ � (resp., � ∘ � ⊆ �),(2) for every � ∈ �, � ∈ � and � ≤ � implies � ∈ �.

If � is both right hyperideal and le� hyperideal of �, then � is called a hyperideal (or two sided hyperideal) of �.

De­nition 6 [7]. A nonempty subset � of (�, ∘, ≤) is called a prime hyperideal of � if the following conditions hold:

(i) � ∘ � ⊆ � ⇒ � ⊆ � or � ⊆ � for any two hyperideals � and � of �;

(ii) if � ∈ � and � ≤ �, then � ∈ � for every � ∈ �.

De­nition 7 [30]. A relation � on an ordered LA-semihy-pergroup � is called a pseudohyperorder if

(1) ≤⊆ �.(2) � is transitive, that is (�, �), (�, �) ∈ � implies (�, �) ∈ �

for all �, �, � ∈ �.(3) � is compatible, that is if (�, �) ∈ � then (� ∘ �, � ∘ �) ∈ �

and (� ∘ �, � ∘ �) ∈ � for all �, �, � ∈ �.

(1)(� ∘ �) ∘ � = (� ∘ �) ∘ �.

(2)(�] = {� ∈ � : � ≤ �, for some � ∈ �}.

3. Hyperfilters in Ordered LA-Semihypergroups

De­nition 8. Let (�, ∘, ≤) be an ordered LA-semihypergroup. An LA-subsemihypergroup � of � is called a le� (resp., right) hyper�lter of � if

(i) for all �, � ∈ �, � ∘ � ⊆ � ⇒ � ∈ � (resp., � ∈ �),(ii) for all � ∈ � and � ∈ �, � ≤ � ⇒ � ∈ �.

De­nition 9. Let (�, ∘, ≤) be an ordered LA-semihypergroup. An LA-subsemihypergroup � of � is called a hyper�lter of �if

(i) for all �, � ∈ �, � ∘ � ⊆ � ⇒ � ∈ � and � ∈ �,(ii) for all � ∈ � and � ∈ �, � ≤ � ⇒ � ∈ �.

Example 10. Consider a set � = {�, �, �} with the following hyperoperation “∘” and the order “≤”:

We give the covering relation “≺” as:

�e �gure of � is shown in Figure 1. �en (�, ∘, ≤) is an ordered LA-semihypergroup. Here {�}, {�}, {�, �} and � are hyper�lters of �.

De­nition 11. Let (�, ∘, ≤) be an ordered LA-semihypergroup. An LA-subsemihypergroup � of � is called a bi-hyper�lter of � if

(i) for all �, � ∈ �, (� ∘ �) ∘ � ⊆ � ⇒ � ∈ �,(ii) for all � ∈ � and � ∈ �, � ≤ � ⇒ � ∈ �.

Example 12. Consider a set � = {�, �, �, �, �} with the following hyperoperation “∘” and the order “≤”:

b c{ , b, c } { , c }

b { b, c } b { , b, c }c { , c } { , b} { , c }≤ := { ( , ), (b, b), (c, ), (c, c )} .

a

aa

aa

aa

a

a aa

a,aa

(3)≺:= {(�, �)}.

c d{(a, a), (a, b), (a, c), (a, d), (b, b), (c, c), (d, d), (e, e),}≤ :=

abcde

abcde

aaaaaa

a a acbde

ab

b

b

c

c

d e

{b, c}{b, c}{a, d}

{a, d}{a, d}{a, d}{a, d}

3Discrete Dynamics in Nature and Society

We give the covering relation “≺” as:

�e �gure of � is shown in Figure 2. �en (�, ∘, ≤) is an ordered LA-semihypergroup. �e bi-hyper�lters of � are {�}, {�, �, �} and �.

De­nition 13. Let (�, ∘, ≤) be an ordered LA-semihypergroup. An LA-subsemihypergroup � of � is called a strong le� (resp., strong right) hyper�lter of � if

(i) for all �, � ∈ �, (� ∘ �) ∩ � ̸= 0 ⇒ � ∈ � (resp., � ∈ �),(ii) for all � ∈ � and � ∈ �, � ≤ � ⇒ � ∈ �.

De­nition 14. Let (�, ∘, ≤) be an ordered LA-semihyp er-group. An LA-subsemihypergroup � of � is called a strong hyper�lter of � if

(i) for all �, � ∈ �, (� ∘ �) ∩ � ̸= 0 ⇒ � ∈ � and � ∈ �,(ii) for all � ∈ � and � ∈ �, � ≤ � ⇒ � ∈ �.

Example 15. Consider a set � = {�, �, �, �, �} with the following hyperoperation “∘” and the order “≤”:

We give the covering relation “≺” as:

�e �gure of � is shown in Figure 3. �en (�, ∘, ≤) is an ordered LA-semihypergroup. �e hyper�lters of � are {�}, {�, �} and �. Here {�} and � are also strong hyper�lters of �. While {�, �} is not a strong hyper�lter of �, because (� ∘ �) ∩ {�, �} = {�} ̸= 0, but � ∉ {�, �} and � ∉ {�, �}.

De­nition 16. Let (�, ∘, ≤) be an ordered LA-semihyper-group. An LA-subsemihypergroup � of � is called a strong bi-hyper�lter of � if

(4)≺= {(�, �), (�, �), (�, �)}.

aaa

{(a, a), (b, b), (c, b), (c, c), (d, b), (d, d), (e, b), (e, e)}

bbbb

{b, c}{b, c}

{b, c}{b, c}

{b, c}

{b, c} {b, c}

{b, c}

bc c c c c c

cc

ccc d

dee

dde

e≤ :=

(5)≺= {(�, �), (�, �), (�, �)}.

(i) for all �, � ∈ �, ((� ∘ �) ∘ �) ∩ � ̸= 0 ⇒ � ∈ �,(ii) for all � ∈ � and � ∈ �, � ≤ � ⇒ � ∈ �.

Example 17. Consider a set � = {�, �, �} with the following hyperoperation “∘” and the order “≤”:

We give the covering relation “≺” as:

�e �gure of � is shown in Figure 4. �en (�, ∘, ≤) is an ordered LA-semihypergroup. �e bi-hyper�lters of � are {�}, {�, �} and �. But � is the only strong bi-hyper�lter of �.

Lemma 18. Let (�, ∘, ≤) be an ordered LA-semihypergroup and � be an LA-subsemihypergroup of �. �en for a le� (resp., right, bi) hyper­lter � of �, either � ∩ � = 0 or � ∩ � is a le� (resp., right, bi) hyper­lter of �.

Proof. Consider � is an LA-subsemihypergroup of �and � is a bi-hyper�lter of �. Suppose that � ∩ � = 0.Now (� ∩ �)2 ⊆ �2 ⊆ � and (� ∩ �)2 ⊆ �2 ⊆ �. �erefore, (� ∩ �)2 ⊆ � ∩ �. So � ∩ � is an LA-subsemihypergroup of �. Suppose that, for any �, � ∈ �, (� ∘ �) ∘ � ⊆ � ∩ �. �erefore, (� ∘ �) ∘ � ⊆ �. As � ∈ � and � is a bi-hyper�lter of �, then � ∈ �. �us, � ∈ � ∩ �. Finally take any � ∈ � ∩ � and � ∈ � such that � ≤ �. As � is a bi-hyper�lter of � with � ∈ � and � ≤ � ∈ �, � ∈ �. �erefore, � ∈ � ∩ �. Hence � ∩ �is a bi-hyper�lter of �. �e other cases can be seen in similar way. ☐

a b ca a { a, b} { a, c }b { a, c } { b, c} cc { a, b} b { b, c}

≤ := { (a, a ), (a, b), (a, c ), (b, b), (c, c )} .

(6)≺= {(�, �), (�, �)}.

a

c

b

Figure 1: Figure of � for Example 10.

b dc

a

e

Figure 2: Figure of � for Example 12.

c ed

b

a

Figure 3: Figure of � for Example 15.

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Lemma 24. Let (�, ∘, ≤) be an ordered LA-semihypergroup and �1, �2 hyper­lters of H. �en �1 ∪ �2 is a hyper­lter of � if and only if �1 ⊆ �2 or �2 ⊆ �1.

Proof. Proof is straightforward. ☐

Theorem 25. Let � be an ordered LA-semihypergroup and �be a nonempty subset of �. �en the following statements are equivalent;

(i) � is a le� hyper�lter of �.(ii) � \ � = 0 or � \ � is a prime right hyperideal of �.

Proof. (i) ⇒ (ii) Assume that � \ � ̸= 0. Let � ∈ � \ � and � ∈ �. If � ∘ � ⊈ � \ �, then � ∘ � ⊆ �. Since � is a le� hyper�lter, so � ∈ � which is not possible. �us, � ∘ � ⊆ � \ �, and so (� \ �) ∘ � ⊂ � \ �. In order to show that � \ � is right hyperideal of �, we need to show that (� \ �] ∘ � ⊆ (� \ �]. Let � ∈ (� \ �] ∘ �, which implies that � ∈ � ∘ � for some � ∈ (� \ �] and � ∈ �. Now consider � ≤ � ∘ � ⊆ � \ �. �us, (� \ �] ∘ � ⊆ (� \ �], which shows that � \ � is right hyperideal of �.

Next we shall show that � \ � is prime. Let � ∘ � ⊆ � \ � for �, � ∈ �. Suppose that � ∉ � \ � and � ∉ � \ �. �en � ∈ � and � ∈ �. Since � is an LA-subsemihypergroup of �, so � ∘ � ⊆ �. It is not possible. �us, � ∈ � \ � and � ∈ � \ �. Hence, � \ �is prime, and so � \ � is a prime right hyperideal.(ii) ⇒ (i) If � \ � = 0, then � = �. �us, in this case, � is

a le� hyper�lter of �. Next assume that � \ � is a prime right hyperideal, then � is an LA-subsemihypergroup of �. Suppose that � ∘ � ⊈ � for �, � ∈ �. �en � ∘ � ⊆ � \ � for �, � ∈ �. Since � \ � is prime, � ∈ � \ � and � ∈ � \ �. It is not pos-sible. �us, � ∘ � ⊆ � and so � is an LA-subsemihypergroup of �. Let � ∘ � ⊆ � for �, � ∈ �. �en, � ∈ �. Indeed, if � ∉ �, then � ∈ � \ �. Since � \ � is prime right hyperideal of �, � ∘ � ⊆ � \ � ∘ � ⊂ � \ �. It is not possible. �us, � ∈ �. �erefore, � is a le� hyper�lter of �. ☐

Theorem 26. Let � be an ordered LA-semihypergroup and �be a nonempty subset of �. �en the following statements are equivalent;

(1) � is a right hyper�lter of �.(2) � \ � = 0 or � \ � is a prime le� hyperideal.

Proof. �e proof is similar to the proof of �eorem 25. ☐

Theorem 27. Let � be an ordered LA-semihypergroup and �be a nonempty subset of �. �en the following statements are equivalent;

(1) � is a hyper�lter of �.(2) � \ � = 0 or � \ � is a prime hyperideal of �.

Proof. Follows directly from �eorems 25 and 26. ☐

Lemma 19. Let (�, ∘, ≤) be an ordered LA-semihypergroup and � be an LA-subsemihypergroup of �. �en for a strong le� (resp., strong right, strong bi) hyper­lter � of �, either � ∩ � = 0 or � ∩ � is a strong le� (resp., strong right, strong bi) hyper­lter of �.

Proof. Proof is straightforward. ☐

Theorem 20. Let (�, ∘, ≤) be an ordered LA-semihypergroup and {��|� ∈ �} a family of le� (resp., right, bi) hyper­lters of �. �en ⋂�∈��� is a le� (resp., right, bi) hyper­lter of � if ⋂�∈��� ̸= 0, where |�| ≥ 2.

Proof. Consider �� is a bi-hyper�lter of � for each � ∈ �. Assume that ⋂�∈��� ̸= 0. Let �, � ∈ ⋂�∈���. �en �, � ∈ �� for all � ∈ �. As, for all � ∈ �, �� is a bi-hyper�lter of �, so � ∘ � ⊆ ��. Hence � ∘ � ⊆ ⋂�∈���, i.e., ⋂�∈��� is an LA-subsemihypergroup of �. Suppose that, for any �, � ∈ �, (� ∘ �) ∘ � ⊆ ⋂�∈���. �erefore, (� ∘ �) ∘ � ⊆ �� for all � ∈ �. As �� is a bi-hyper�lter of �, then � ∈ �� for all � ∈ �. �us, � ∈ ⋂�∈���. Finally take any � ∈ ⋂�∈���and � ∈ � such that � ≤ �. �erefore, � ∈ �� for all � ∈ �. As ��is a bi-hyper�lter of � with � ∈ �� and � ≤ � ∈ �, this implies � ∈ �� for all � ∈ �. �erefore, � ∈ ⋂�∈���. Hence ⋂�∈��� is a bi-hyper�lter of �. �e other cases can be seen in similar way. ☐

Theorem 21. Let (�, ∘, ≤) be an ordered LA-semihypergroup and {��|� ∈ �} a family of strong le� (resp., strong right, strong bi) hyper­lters of �. �en ⋂�∈��� is a strong le� (resp., strong right, strong bi) hyper­lter of � if ⋂�∈��� ̸= 0, where |�| ≥ 2.

Proof. Proof is straightforward. ☐

Remark 22. Generally, the union of two hyper�lters of an ordered LA-semihypergroup � need not be a hyper�lter of �.

�e following example shows that, the union of two hyper�lters of an ordered LA-semihypergroup � is not a hyper�lter of �.

Example 23. Consider Example 1. One can easily see that �1 = {�} and �2 = {�} are both hyper�lters of �. But �1 ∪ �2 = {�, �} is not a hyper�lter of �, because (�1 ∪ �2)

2 ⊈ (�1 ∪ �2), i.e., �1 ∪ �2 is not an LA-subsemihyper-group of �.

b c

a

Figure 4: Figure of � for Example 17.

5Discrete Dynamics in Nature and Society

� ∈ ��(�). �us, � ∘ � ⊆ ��(�) ∘ ��(�) ⊆ �. Since � is a le� hyper�lter of �, so we have � ∈ �, which is a contradiction. Now, let � ∈ �−(�) and � ∈ � such that � ≤ �. �en ��(�) ⊆ � and ���. �is implies that ��(�) = ��(�). Since ��(�) ⊆ �, so ��(�) ⊆ �. �us, � ∈ �−(�). �us, �−(�)is a le� hyper�lter of �. �e case for right hyper�lter and hyper�lter can be proved in a similar way. ☐

De­nition 33. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-upper rough le� hyper�lter (resp., right hyper�lter, hyper�lter) of � if �+(�) is a le� hyper�lter (resp., right hyper�lter, hyper�lter) of �.

Theorem 34. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a le� hyper­lter (resp., right hyper­lter, hyper­lter) of �. �en �+(�) is a le� hyper­lter (resp., right hyper­lter, hyper­lter) of �.

Proof. Let � be a le� hyper�lter of �. Since � is an LA-subsemihypergroup of �, then by �eorem 30(1), �+(�) is an LA-subsemihypergroup of �. Let � ∘ � ⊆ �+(�) for some �, � ∈ �. �en (��(�) ∘ ��(�)) ∩ � = ��(� ∘ �) ∩ � ̸= 0, so there exist � ∈ ��(�) and � ∈ ��(�) such that � ∘ � ⊆ �. Since � is le� hyper�lter of �, so we have � ∈ �. �us, ��(�) ∩ � ̸= 0 which implies that � ∈ �+(�). Now, let � ∈ �+(�) and � ∈ � such that � ≤ �. �en ��(�) ∩ � ̸= 0 and ���. �is implies that ��(�) = ��(�). Since ��(�) ∩ � ̸= 0, so ��(�) ∩ � ̸= 0. �us, � ∈ �+(�). �us, �+(�) is a le� hyper�lter of �, that is, � is a �-upper rough le� hyper�lter of �. �e case for right hyper�lter and hyper�lter can be proved in a similar way. ☐

�e converse of �eorems 32 and 34 is not true in general.

Example 35. Consider a set � = {�, �, �, �, �} with the following hyperoperation “∘” and the order “≤”:

ab

a b c d ea b c d ec d e

c b { d, e }{ d, e }{ d, e } { d, e }{ d, e }

{ d, e }{ d, e } { d, e }{ d, e }

{ d, e }{ d, e }d { d, e } e

d d ee e e e e e

≤ := { (a, a), (b, b), (c, c), (d, d), (d, e), (e, e)}.

4. Rough Hyperfilters in Ordered LA-Semihypergroups

In this section, we will study some results on hyper�lters of ordered LA-semihypergroups in terms of rough sets which are based on pseudohyperorder relation.

De­nition 28 [30]. Let � be a nonempty set and � be a binary relation on �. By P(�) we mean the power set of �. For all � ⊆ �, we de�ne �− and �+ :P(�) →P(�) by

and

where ��(�) = {� ∈ � : ���}. �−(�) and �+(�) are called the lower approximation and the upper approximation operations, respectively.

De­nition 29 [30, De­nition 4.1]. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-upper (resp., �-lower) rough LA-subsemihypergroup of � if �+(�) (resp., �−(�)) is an LA-subsemihypergroup of �.

Theorem 30 [30, �eorem 4.3]. Let � be a pseudohyperorder on an ordered LA-semihypergroup � and � an LA-subsemihypergroup of �. �en

(1) �+(�) is an LA-subsemihypergroup of �.(2) if � is complete, then �−(�) is, if it is nonempty, an

LA-subsemihypergroup of �.

De­nition 31. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-lower rough le� hyper�lter (resp., right hyper�lter, hyper�lter) of � if �−(�) is a le� hyper�lter (resp., right hyper�lter, hyper�lter) of �.

Theorem 32. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a le� hyper­lter (resp., right hyper­lter, hyper­lter) of �. �en �−(�) is, if it is nonempty, a le� hyper­lter (resp., right hyper­lter, hyper­lter) of �.

Proof. Let � be a le� hyper�lter of �. Since � is an LA-subsemihypergroup of �, then by �eorem 30(2), �−(�)is an LA-subsemihypergroup of �. Let � ∘ � ⊆ �−(�) for some �, � ∈ �. �en ��(�) ∘ ��(�) = ��(� ∘ �) ⊆ �. We suppose that �−(�) is not a le� hyper�lter of �. �en there exist �, � ∈ � such that � ∘ � ⊆ �−(�) but � ∉ �−(�). �us, ��(�) ⊈ �. �en there exist � ∈ ��(�) such that � ∉ � and

(7)�−(�) = {� ∈ � : ∀�, ���⇒ � ∈ �}= {� ∈ � : ��(�) ⊆ �},

(8)�+(�) = {� ∈ � : ∃� ∈ �, such that ���}= {� ∈ � : ��(�) ∩ � ̸= 0},

e

d

ba c

Figure 5: Figure of � for Example 35.

Discrete Dynamics in Nature and Society6

of �, so we have � ∈ �. �us, ��(�) ∩ � ̸= 0 which implies that � ∈ �+(�). Now, let � ∈ �+(�) and � ∈ � such that � ≤ �. �en ��(�) ∩ � ̸= 0 and ���. �is implies that ��(�) = ��(�). Since ��(�) ∩ � ̸= 0, so ��(�) ∩ � ̸= 0.�us, � ∈ �+(�). �us, �+(�) is a bi-hyper�lter of �, that is, �is a �-upper rough bi-hyper�lter of �. ☐

De­nition 39. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-lower rough strong le� hyper�lter (resp., strong right hyper�lter, strong hyper�lter) of � if �−(�) is a strong le� hyper�lter (resp., strong right hyper�lter, strong hyper�lter) of �.

Theorem 40. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a strong le� hyper­lter (resp., strong right hyper­lter, strong hyper­lter) of �. �en �−(�) is, if it is nonempty, a strong le� hyper­lter (resp., strong right hyper­lter, strong hyper­lter) of �.

Proof. Let � be a strong hyper�lter of �. Since � is an LA-subsemihypergroup of �, then by �eorem 30(2), �−(�) is an LA-subsemihypergroup of �. Let (� ∘ �) ∩ �−(�) ̸= 0 for some �, � ∈ �. For � ∈ (� ∘ �) ∩ �−(�), we have � ∈ � ∘ �and � ∈ �−(�). �is implies that ��(�) ⊆ ��(� ∘ �) =��(�) ∘ ��(�) and ��(�) ⊆ �. �is implies that ��(�) ⊆ (��(�) ∘ ��(�)) ∩ �. �us (��(�) ∘ ��(�)) ∩ � ̸= 0. We suppose that �−(�) is not a strong hyper�lter of �. �en there exist �, � ∈ � such that (� ∘ �) ∩ �−(�) ̸= 0 but � ∉ �−(�) and � ∉ �−(�). �us, ��(�) ⊈ � and ��(�) ⊈ �. �en there exist � ∈ ��(�) and � ∈ ��(�) such that � ∉ �and � ∉ �. �us, (� ∘ �) ∩ � ⊆ (��(�) ∘ ��(�)) ∩ � ̸= 0. Since � is a strong hyper�lter of �, so we have � ∈ � and � ∈ �, which is a contradiction. Now, let � ∈ �−(�) and � ∈ �such that � ≤ �. �en ��(�) ⊆ � and ���. �is implies that ��(�) = ��(�). Since ��(�) ⊆ �, so ��(�) ⊆ �. �us, � ∈ �−(�). Hence �−(�) is a strong hyper�lter of �. �e case for strong le� hyper�lter and strong right hyper�lter can be proved in a similar way. ☐

De­nition 41. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-upper rough strong le� hyper�lter (resp., strong right hyper�lter, strong hyper�lter) of � if �+(�) is a strong le� hyper�lter (resp., strong right hyper�lter, strong hyper�lter) of �.

Theorem 42. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a strong le� hyper­lter (resp., strong right hyper­lter, strong hyper­lter) of �. �en �+(�) is a strong le� hyper­lter (resp., strong right hyper­lter, strong hyper­lter) of �.

Proof. Let � be a strong hyper�lter of �. Since � is an LA-subsemihypergroup of �, then by �eorem 30(1), �+(�) is an

We give the covering relation “≺” as:

�e �gure of � is shown in Figure 5. �en (�, ∘, ≤) is an ordered LA-semihypergroup. Now let

be a complete pseudohyperorder on �, such that

Now for {�, �, �} ⊆ �,

It is clear that �−({�, �, �}) and �+({�, �, �}) are both hyper�l-ters of �, but {�, �, �} is not a hyper�lter of �.

De­nition 35. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-lower rough bi-hyper�lter of � if �−(�) is a bi-hyper�lter of �.

Theorem 36. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a bi-hyper­lter of �. �en �−(�) is, if it is nonempty, a bi-hyper­lter of �.

Proof. Let � be a bi-hyper�lter of �. Since � is an LA-subse mih-ypergroup of �, then by �eorem 30(2), �−(�) is an LA-subsemi-hypergroup of �. Let (� ∘ �) ∘ � ⊆ �−(�) for some �, � ∈ �. �en (��(�) ∘ ��(�)) ∘ ��(�) = ��((� ∘ �) ∘ �) ⊆ �. We suppose that �−(�) is not a bi-hyper�lter of �. �en there exist �, � ∈ � such that (� ∘ �) ∘ � ⊆ �−(�), but � ∉ �−(�). �us, ��(�) ⊈ �. �en there exist � ∈ ��(�) such that � ∉ � and � ∈ ��(�). �us, (� ∘ �) ∘ � ⊆ (��(�) ∘ ��(�)) ∘ ��(�) ⊆ �. Since � is a bi-hyper�lter of �, so we have � ∈ �, which is a contradiction. Now, let � ∈ �−(�) and � ∈ � such that � ≤ �. �en ��(�) ⊆ � and ���. �is implies that ��(�) = ��(�). Since ��(�) ⊆ �, so ��(�) ⊆ �. �us � ∈ �−(�). Hence �−(�) is a bi-hyper�lter of �. ☐

De­nition 37. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-upper rough bi-hyper�lter of � if �+(�) is a bi-hyper�lter of �.

Theorem 38. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a bi-hyper­lter of �. �en �+(�) is a bi-hyper­lter of �.

Proof. Let � be a bi-hyper�lter of �. Since � is an LA-subsemihypergroup of �, then by �eorem 30(1), �+(�) is an LA-subsemihypergroup of �. Let (� ∘ �) ∘ � ⊆�+(�) for some �, � ∈ �. �en ((��(�) ∘ ��(�)) ∘ ��(�))∩� = ��((� ∘ �) ∘ �) ∩ � ̸= 0, so there exist � ∈ ��(�) and � ∈ ��(�) such that (� ∘ �) ∘ � ⊆ �. Since � is bi-hyper�lter

(9)≺= {(�, �)}.

(10)� = {(�, �), (�, �), (�, �), (�, �), (�, �), (�, �), (�, �), (�, �), (�, �)}

(11)��(�) = {�}, ��(�) = ��(�) = {�, �} and ��(�) = ��(�) = {�, �}.

(12)�−({�, �, �}) = {�} and �+({�, �, �}) = {�, �, �, �, �}.

7Discrete Dynamics in Nature and Society

Theorem 45. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a strong bi-hyper­lter of �. �en �−(�) is, if it is nonempty, a strong bi-hyper­lter of �.

Proof. �e proof is similar to the proof of �eorem 40. ☐

De­nition 46. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-upper rough strong bi-hyper�lter of � if �+(�) is a strong bi-hyper�lter of �.

Theorem 47. Let � be a complete pseudohyperorder on an ordered LA-semihypergroup � and � a strong bi-hyper­lter of �. �en �+(�) is a strong bi-hyper­lter of �.

Proof. �e proof is similar to the proof of �eorem 42. ☐

Data Availability

No data were used to support this study.

Ethical Approval

�is article does not contain any studies with human partici-pants or animals performed by any of the authors.

Conflicts of Interest

�e authors of this paper Ferdaous Bouaziz and Naveed Yaqoob declare that they have no con·icts of interest.

Acknowledgments

�e authors gratefully acknowledge the support of the Deanship of Scienti�c Research, Qassim University, for this research under the Grant number (3619-cosabu-2018-1-14-S) during the academic year 1439-1440 AH/2018 AD.

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LA-subsemihypergroup of �. Let (� ∘ �) ∩ �+(�) ̸= 0 for some �, � ∈ �. �en (��(�) ∘ ��(�)) ∩ � = ��(� ∘ �) ∩ � ̸= 0, so there exist � ∈ ��(�) and � ∈ ��(�) such that (� ∘ �) ∩ � ̸= 0. Since � is strong hyper�lter of �, so we have � ∈ � and � ∈ �. �us, ��(�) ∩ � ̸= 0 and ��(�) ∩ � ̸= 0, which implies that � ∈ �+(�) and � ∈ �+(�). Now, let � ∈ �+(�) and � ∈ � such that � ≤ �. �en ��(�) ∩ � ̸= 0and ���. �is implies that ��(�) = ��(�). Since ��(�) ∩ � ̸= 0, so ��(�) ∩ � ̸= 0. �us, � ∈ �+(�). �us, �+(�) is a strong hyper�lter of �, that is, � is a �-upper rough strong hyper�lter of �. �e case for strong le� hyper�lter and strong right hyper�lter can be proved in a similar way. ☐

�e converse of �eorems 39 and 41 is not true in general.

Example 43. Consider a set � = {�, �, �, �, �} with the following hyperoperation “∘” and the order “≤”:

We give the covering relation “≺” as:

�e �gure of � is shown in Figure 6. �en (�, ∘, ≤) is an ordered LA-semihypergroup. Now let � be a complete pseu-dohyperorder on �, such that

Now for {�, �, �} ⊆ �,

It is clear that �−({�, �, �}) and �+({�, �, �}) are both strong hyper�lters of � but {�, �, �} is not a hyper�lter of �.

De­nition 44. Let � be a pseudohyperorder on an ordered LA-semihypergroup �. �en a nonempty subset � of � is called a �-lower rough strong bi-hyper�lter of � if �−(�) is a strong bi-hyper�lter of �.

a b c d ea b b e e eb { a, b} { a, b} e e ec e e { c, d} d ed e e c { c, d} ee e e e e e

≤:= { (a, a), (a, b), (b, b), (c, c), (d, d), (e, c), (e, d), (e, e)}.

(13)≺= {(�, �), (�, �), (�, �)}.

(14)��(�) = ��(�) = {�, �} and ��(�) = ��(�) = ��(�) = {�, �, �}.

(15)�−({�, �, �}) = {�, �} and �+({�, �, �}) = {�, �, �, �, �}.

b

a e

c d

Figure 6: Figure of � for Example 42.

Discrete Dynamics in Nature and Society8

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