Rough Hyperfilters in Po-LA-Semihypergroups

Bouaziz, Ferdaous; Yaqoob, Naveed

doi:https://doi.org/10.1155/2019/8326124

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Data Availability Ethical Approval Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 8326124 | https://doi.org/10.1155/2019/8326124

Rough Hyperfilters in Po-LA-Semihypergroups

Ferdaous Bouaziz¹and Naveed Yaqoob²

Academic Editor: Xiaohua Ding

Received04 Jul 2019

Revised30 Aug 2019

Accepted13 Sept 2019

Published29 Dec 2019

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.

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 -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], -semihypergroups [20], hyperrings [21], Hv-groups [22], Hv-modules [23], -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 hypergroupoid is said to be an LA-semihypergroup if for all ,

Definition 2 [7]. Let be a nonempty set and “” be an ordered relation on . Then 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 .

Definition 3 [7]. If is a nonempty subset of , then is the subset of defined as follows:

Definition 4 [7]. A nonempty subset of an ordered LA-semihypergroup is called an LA-subsemihypergroup of if .

Definition 5 [7]. A nonempty subset of an ordered LA-semihypergroup is called a right (resp., left) hyperideal of if(1) ,(2)for every , and implies .If is both right hyperideal and left hyperideal of , then is called a hyperideal (or two sided hyperideal) of .

Definition 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 .

Definition 7 [30]. A relation on an ordered LA-semihypergroup is called a pseudohyperorder if(1).(2) is transitive, that is implies for all (3) is compatible, that is if then and for all

3. Hyperfilters in Ordered LA-Semihypergroups

Definition 8. Let be an ordered LA-semihypergroup. An LA-subsemihypergroup of is called a left (resp., right) hyperfilter of if(i)for all , (resp., ),(ii)for all and ,

Definition 9. Let be an ordered LA-semihypergroup. An LA-subsemihypergroup of is called a hyperfilter 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:The figure of is shown in Figure 1. Then is an ordered LA-semihypergroup. Here and are hyperfilters of .

Definition 11. Let be an ordered LA-semihypergroup. An LA-subsemihypergroup of is called a bi-hyperfilter of if(i)for all , ,(ii)for all and ,

Example 12. Consider a set with the following hyperoperation “” and the order “”:

We give the covering relation “” as:The figure of is shown in Figure 2. Then is an ordered LA-semihypergroup. The bi-hyperfilters of are , and .

Definition 13. Let be an ordered LA-semihypergroup. An LA-subsemihypergroup of is called a strong left (resp., strong right) hyperfilter of if(i)for all , (resp., ),(ii)for all and ,

Definition 14. Let be an ordered LA-semihypergroup. An LA-subsemihypergroup of is called a strong hyperfilter of if(i)for all , and ,(ii)for all and ,

Example 15. Consider a set with the following hyperoperation “” and the order “”:

We give the covering relation “” as:The figure of is shown in Figure 3. Then is an ordered LA-semihypergroup. The hyperfilters of are , and . Here and are also strong hyperfilters of . While is not a strong hyperfilter of , because , but and

Definition 16. Let be an ordered LA-semihypergroup. An LA-subsemihypergroup of is called a strong bi-hyperfilter of if(i)for all , ,(ii)for all and ,

Example 17. Consider a set with the following hyperoperation “” and the order “”:

We give the covering relation “” as:The figure of is shown in Figure 4. Then is an ordered LA-semihypergroup. The bi-hyperfilters of are , and . But is the only strong bi-hyperfilter of .

Lemma 18. Let be an ordered LA-semihypergroup and be an LA-subsemihypergroup of . Then for a left (resp., right, bi) hyperfilter of , either or is a left (resp., right, bi) hyperfilter of .

Proof. Consider is an LA-subsemihypergroup of and is a bi-hyperfilter of . Suppose that Now and . Therefore, . So is an LA-subsemihypergroup of . Suppose that, for any , . Therefore, . As and is a bi-hyperfilter of , then . Thus, . Finally take any and such that . As is a bi-hyperfilter of with and , . Therefore, . Hence is a bi-hyperfilter of . The other cases can be seen in similar way.

Lemma 19. Let be an ordered LA-semihypergroup and be an LA-subsemihypergroup of . Then for a strong left (resp., strong right, strong bi) hyperfilter of , either or is a strong left (resp., strong right, strong bi) hyperfilter of .

Proof. Proof is straightforward.

Theorem 20. Let be an ordered LA-semihypergroup and a family of left (resp., right, bi) hyperfilters of . Then is a left (resp., right, bi) hyperfilter of if , where .

Proof. Consider is a bi-hyperfilter of for each . Assume that Let Then for all . As, for all , is a bi-hyperfilter of , so . Hence , i.e., is an LA-subsemihypergroup of . Suppose that, for any , Therefore, for all . As is a bi-hyperfilter of , then for all . Thus, . Finally take any and such that . Therefore, for all . As is a bi-hyperfilter of with and , this implies for all . Therefore, Hence is a bi-hyperfilter of . The other cases can be seen in similar way.

Theorem 21. Let be an ordered LA-semihypergroup and a family of strong left (resp., strong right, strong bi) hyperfilters of . Then is a strong left (resp., strong right, strong bi) hyperfilter of if , where .

Proof. Proof is straightforward.

Remark 22. Generally, the union of two hyperfilters of an ordered LA-semihypergroup need not be a hyperfilter of .

The following example shows that, the union of two hyperfilters of an ordered LA-semihypergroup is not a hyperfilter of .

Example 23. Consider Example 1. One can easily see that and are both hyperfilters of . But is not a hyperfilter of , because , i.e., is not an LA-subsemihypergroup of .

Lemma 24. Let be an ordered LA-semihypergroup and , hyperfilters of H. Then is a hyperfilter of if and only if or .

Proof. Proof is straightforward.

Theorem 25. Let be an ordered LA-semihypergroup and be a nonempty subset of . Then the following statements are equivalent;(i) is a left hyperfilter of .(ii) or is a prime right hyperideal of .

Proof. (i) (ii) Assume that Let and . If , then . Since is a left hyperfilter, so which is not possible. Thus, , 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 . Thus, , which shows that is right hyperideal of .
Next we shall show that is prime. Let for . Suppose that and . Then and . Since is an LA-subsemihypergroup of , so . It is not possible. Thus, and . Hence, is prime, and so is a prime right hyperideal.
(ii) (i) If , then . Thus, in this case, is a left hyperfilter of . Next assume that is a prime right hyperideal, then is an LA-subsemihypergroup of . Suppose that for . Then for . Since is prime, and . It is not possible. Thus, and so is an LA-subsemihypergroup of . Let for . Then, . Indeed, if , then . Since is prime right hyperideal of , . It is not possible. Thus, . Therefore, is a left hyperfilter of .

Theorem 26. Let be an ordered LA-semihypergroup and be a nonempty subset of . Then the following statements are equivalent;(1) is a right hyperfilter of .(2) or is a prime left hyperideal.

Proof. The proof is similar to the proof of Theorem 25.

Theorem 27. Let be an ordered LA-semihypergroup and be a nonempty subset of . Then the following statements are equivalent;(1) is a hyperfilter of .(2) or is a prime hyperideal of .

Proof. Follows directly from Theorems 25 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]. Let be a nonempty set and be a binary relation on . By we mean the power set of . For all , we define and by and where . and are called the lower approximation and the upper approximation operations, respectively.

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

Theorem 30 [30, Theorem 4.3]. Let be a pseudohyperorder on an ordered LA-semihypergroup and an LA-subsemihypergroup of . Then(1) is an LA-subsemihypergroup of .(2)if is complete, then is, if it is nonempty, an LA-subsemihypergroup of .

Definition 31. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -lower rough left hyperfilter (resp., right hyperfilter, hyperfilter) of if is a left hyperfilter (resp., right hyperfilter, hyperfilter) of .

Theorem 32. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a left hyperfilter (resp., right hyperfilter, hyperfilter) of . Then is, if it is nonempty, a left hyperfilter (resp., right hyperfilter, hyperfilter) of .

Proof. Let be a left hyperfilter of . Since is an LA-subsemihypergroup of , then by Theorem 30(2), is an LA-subsemihypergroup of . Let for some . Then . We suppose that is not a left hyperfilter of . Then there exist such that but . Thus, . Then there exist such that and . Thus, . Since is a left hyperfilter of , so we have , which is a contradiction. Now, let and such that . Then and . This implies that . Since , so . Thus, . Thus, is a left hyperfilter of . The case for right hyperfilter and hyperfilter can be proved in a similar way.

Definition 33. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -upper rough left hyperfilter (resp., right hyperfilter, hyperfilter) of if is a left hyperfilter (resp., right hyperfilter, hyperfilter) of .

Theorem 34. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a left hyperfilter (resp., right hyperfilter, hyperfilter) of . Then is a left hyperfilter (resp., right hyperfilter, hyperfilter) of .

Proof. Let be a left hyperfilter of . Since is an LA-subsemihypergroup of , then by Theorem 30(1), is an LA-subsemihypergroup of . Let for some . Then , so there exist and such that . Since is left hyperfilter of , so we have . Thus, which implies that . Now, let and such that . Then and . This implies that . Since , so . Thus, . Thus, is a left hyperfilter of , that is, is a -upper rough left hyperfilter of . The case for right hyperfilter and hyperfilter can be proved in a similar way.

The converse of Theorems 32 and 34 is not true in general.

Example 35. Consider a set with the following hyperoperation “” and the order “”:

We give the covering relation “” as:The figure of is shown in Figure 5. Then is an ordered LA-semihypergroup. Now letbe a complete pseudohyperorder on , such thatNow for ,It is clear that and are both hyperfilters of , but is not a hyperfilter of .

Definition 35. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -lower rough bi-hyperfilter of if is a bi-hyperfilter of .

Theorem 36. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a bi-hyperfilter of . Then is, if it is nonempty, a bi-hyperfilter of .

Proof. Let be a bi-hyperfilter of . Since is an LA-subsemihypergroup of , then by Theorem 30(2), is an LA-subsemihypergroup of . Let for some . Then . We suppose that is not a bi-hyperfilter of . Then there exist such that , but . Thus, . Then there exist such that and . Thus, . Since is a bi-hyperfilter of , so we have , which is a contradiction. Now, let and such that . Then and . This implies that . Since , so . Thus . Hence is a bi-hyperfilter of .

Definition 37. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -upper rough bi-hyperfilter of if is a bi-hyperfilter of .

Theorem 38. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a bi-hyperfilter of . Then is a bi-hyperfilter of .

Proof. Let be a bi-hyperfilter of . Since is an LA-subsemihypergroup of , then by Theorem 30(1), is an LA-subsemihypergroup of . Let for some . Then , so there exist and such that . Since is bi-hyperfilter of , so we have . Thus, which implies that . Now, let and such that . Then and . This implies that . Since , so Thus, . Thus, is a bi-hyperfilter of , that is, is a -upper rough bi-hyperfilter of .

Definition 39. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -lower rough strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of if is a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of .

Theorem 40. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of . Then is, if it is nonempty, a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of .

Proof. Let be a strong hyperfilter of . Since is an LA-subsemihypergroup of , then by Theorem 30(2), is an LA-subsemihypergroup of . Let for some . For , we have and . This implies that and . This implies that . Thus . We suppose that is not a strong hyperfilter of . Then there exist such that but and . Thus, and . Then there exist and such that and . Thus, . Since is a strong hyperfilter of , so we have and , which is a contradiction. Now, let and such that . Then and . This implies that . Since , so . Thus, . Hence is a strong hyperfilter of . The case for strong left hyperfilter and strong right hyperfilter can be proved in a similar way.

Definition 41. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -upper rough strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of if is a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of .

Theorem 42. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of . Then is a strong left hyperfilter (resp., strong right hyperfilter, strong hyperfilter) of .

Proof. Let be a strong hyperfilter of . Since is an LA-subsemihypergroup of , then by Theorem 30(1), is an LA-subsemihypergroup of . Let for some . Then , so there exist and such that . Since is strong hyperfilter of , so we have and . Thus, and , which implies that and . Now, let and such that . Then and . This implies that . Since , so . Thus, . Thus, is a strong hyperfilter of , that is, is a -upper rough strong hyperfilter of . The case for strong left hyperfilter and strong right hyperfilter can be proved in a similar way.

The converse of Theorems 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:The figure of is shown in Figure 6. Then is an ordered LA-semihypergroup. Now let be a complete pseudohyperorder on , such thatNow for ,It is clear that and are both strong hyperfilters of but is not a hyperfilter of .

Definition 44. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -lower rough strong bi-hyperfilter of if is a strong bi-hyperfilter of .

Theorem 45. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a strong bi-hyperfilter of . Then is, if it is nonempty, a strong bi-hyperfilter of .

Proof. The proof is similar to the proof of Theorem 40.

Definition 46. Let be a pseudohyperorder on an ordered LA-semihypergroup . Then a nonempty subset of is called a -upper rough strong bi-hyperfilter of if is a strong bi-hyperfilter of .

Theorem 47. Let be a complete pseudohyperorder on an ordered LA-semihypergroup and a strong bi-hyperfilter of . Then is a strong bi-hyperfilter of .

Proof. The proof is similar to the proof of Theorem 42.

Data Availability

No data were used to support this study.

Ethical Approval

This article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest

The authors of this paper Ferdaous Bouaziz and Naveed Yaqoob declare that they have no conflicts of interest.

Acknowledgments

The authors gratefully acknowledge the support of the Deanship of Scientific 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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Copyright © 2019 Ferdaous Bouaziz and Naveed Yaqoob. This 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.

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