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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 9494072, 13 pages
https://doi.org/10.1155/2018/9494072
Research Article

On -Fuzzy Hyperideals in Ordered LA-Semihypergroups

1Department of Mathematics, Hazara University, Mansehra, KP, Pakistan
2Department of Mathematics, College of Science Al-Zulfi, Majmaah University, Al-Zulfi, Saudi Arabia
3High Institute of Engineering and Technology, King Marioutt, P.O. Box 3135, Egypt

Correspondence should be addressed to Naveed Yaqoob; moc.liamy@booqayan

Received 25 January 2018; Accepted 17 May 2018; Published 8 July 2018

Academic Editor: Rigoberto Medina

Copyright © 2018 Muhammad Azhar et al. 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.

Abstract

The concept of -fuzzy hyperideal of an ordered LA-semihypergroup is introduced by the ordered fuzzy points, and related properties are investigated. We study the relations between different -fuzzy hyperideal of an ordered LA-semihypergroup. Furthermore, we study some results on homomorphisms of -fuzzy hyperideals.

1. Introduction

Hyperstructures represent a natural extension of classical algebraic structures and they were introduced by the French mathematician Marty [1]. Since then, several authors continued their researches in this direction. Nowadays hyperstructures are widely studied from theoretical point of view and for their applications in many subjects of pure and applied mathematics. In a classical algebraic structure, the composition of two elements is an element, while, in an algebraic hyperstructure, the composition of two elements is a set. Some basic definitions and theorems about hyperstructures can be found in [2, 3]. The concept of a semihypergroup is a generalization of the concept of a semigroup. Many authors studied different aspects of semihypergroups, for instance, Bonansinga and Corsini [4], Corsini [5] Davvaz [6], Davvaz et al. [7], Davvaz and Poursalavati [8], Gutan [9], Hasankhani [10], Hila et al. [11], and Onipchuk [12]. Recently, Hila and Dine [13] introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semihypergroups. Yaqoob, Corsini, and Yousafzai [14] extended the work of Hila and Dine and characterized intraregular left almost semihypergroups by their hyperideals using pure left identities. Other results on LA-semihypergroups can be found in [15, 16].

The concept of ordered semihypergroup was studied by Heidari and Davvaz in [17], where they used a binary relation “” on semihypergroup such that the binary relation is a partial order and the structure is known as ordered semihypergroup. There are several authors who study the ordering of hyperstructures, for instance, Bakhshi and Borzooei [18], Chvalina [19], Hoskova [20], Kondo and Lekkoksung [21], and Novak [22]. The ordering in LA-semihypergroups was introduced by Yaqoob and Gulistan [23].

In 1965, the concept of fuzzy sets was first proposed by Zadeh [24], which has a wide range of applications in various fields such as computer engineering, artificial intelligence, control engineering, operation research, management science, robotics, and many more. Many papers on fuzzy sets have been published, showing the importance and their applications to set theory, group theory, real analysis, measure theory, and topology etc. There are several authors who applied the concept of fuzzy sets to algebraic hyperstructures. Ameri and others studied fuzzy sets in hypervector spaces [25], -hyperrings [26], and polygroups [27]. Corsini et al. [28] studied semisipmle semihypergroups in terms of hyperideals and fuzzy hyperideals. Cristea [29] introduced hyperstructures and fuzzy sets endowed with two membership functions. In [30, 31], Davvaz introduced the concept of fuzzy hyperideals in a semihypergroup and Hv-semigroup. Recently in [32], Davvaz and Leoreanu-Fotea studied the structure of fuzzy -hyperideals in -semihypergroups. Also see [33, 34]. Fuzzy ordered hyperstructures have been considered by some researchers, for instance, Pibaljommee et al. [35, 36], Tang et al. [3740], Tipachot and Pibaljommee [41].

Murali [42] defined the concept of belongingness of a fuzzy point to a fuzzy subset under a natural equivalence on a fuzzy subset. In [43], the idea of quasi-coincidence of a fuzzy point with a fuzzy set is defined. The concept of a -fuzzy subgroup was first considered by Bhakat and Das in [44, 45] by using the “belongs to” relation and “quasi coincident with” relation between a fuzzy point and a fuzzy subgroup, where and Jun et al. [46] gave the concept of -fuzzy ordered semigroups. Moreover, -fuzzy ideals, -fuzzy quasi-ideals, and -fuzzy bi-ideals of a semigroup are defined in [47]. Azizpour and Talebi characterized semihypergroups by -fuzzy interior hyperideals [48, 49]. Huang [50] provided characterizations of semihyperrings by their -fuzzy hyperideals. Shabir and Mahmood characterized semihypergroups by -fuzzy hyperideals [51] and also by -fuzzy hyperideals [52].

Motivated by the work of Shabir et al. [47, 51], we applied the concept of -fuzzy sets to LA-semihypergroups. In this paper, we introduce and study several types of -fuzzy hyperideals in an ordered LA-semihypergroup. Further, we study some results on homomorphisms of -fuzzy hyperideals.

2. Preliminaries and Basic Definitions

In this section, we provide some basic definitions needed for our further work.

Let be a nonempty set. Then the map is called hyperoperation or join operation on the set , where denotes the set of all nonempty subsets of . A hypergroupoid is a set together with a (binary) hyperoperation. For any nonempty subsets of , we denote Instead of and , we write and , respectively.

Recently, in [13, 14] authors introduced the notion of LA-semihypergroups as a generalization of semigroups, semihypergroups, and LA-semihypergroups. A hypergroupoid is called an LA-semihypergroup if for every , we have The law is called a left invertive law. An element is called a left identity (resp., pure left identity) if for all , (resp., ). In an LA-semihypergroup, the medial law holds for all . An LA-semihypergroup may or may not contain a left identity and pure left identity. In an LA-semihypergroup with pure left identity, the paramedial law holds for all If an LA-semihypergroup contains a pure left identity, then by using medial law, we get for all .

Definition 1 (see [23]). Let be a nonempty set and be an ordered relation on . The triplet is called an ordered LA-semihypergroup if the following conditions are satisfied.
   is an LA-semihypergroup.
   is a partially ordered set.
For every , implies and , where means that for there exist such that

Definition 2 (see [23]). If is an ordered LA-semihypergroup and , then is the subset of defined as follows: If and is a nonempty subset of , then

Definition 3 (see [23]). A nonempty subset of an ordered LA-semihypergroup is called an LA-subsemihypergroup of if

Definition 4 (see [23]). A nonempty subset of an ordered LA-semihypergroup is called a right (resp., left) hyperideal of if
   (resp., ),
for every , and implies .

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

Definition 5 (see [23]). A nonempty subset of an ordered LA-semihypergroup is called an interior hyperideal of if
  ,
for every , and implies .

Definition 6 (see [23]). A nonempty subset of an ordered LA-semihypergroup is called a generalized bi-hyperideal of if
  ,
for every , and implies .

If is both an LA-subsemihypergroup and a generalized bi-hyperideal of , then is called a bi-hyperideal of .

Definition 7. Let be an ordered LA-semihypergroup, and Then is said to be regular element of if there exists an element such that , or equivalently If every element of is regular then is said to be a regular ordered LA-semihypergroup.

Now, we give some fuzzy logic concepts.

A fuzzy subset of a universe is a function from into the unit closed interval , i.e., (see [24]). A fuzzy subset in a universe of the form is said to be a fuzzy point with support and value and is denoted by For a fuzzy point and a fuzzy set in a set , Pu and Liu [43] introduced the symbol , where , which means that

(i) if ,

(ii) if ,

(iii) if or ,

(iv) if and ,

(v) if does not hold.

For any two fuzzy subsets and of , means that, for all , The symbols and will mean the following fuzzy subsets: for all

The product of any fuzzy subsets and of is defined by For , the characteristic function of is a fuzzy subset of , defined by For any fuzzy subset of and for any , the set is called a level subset of

Definition 8 (see [53]). Let be an ordered LA-semihypergroup. A fuzzy subset is called fuzzy LA-subsemihypergroup of if the following assertion are satisfied.
(i) ,
(ii) if implies , for every

Definition 9 (see [53]). Let be an ordered LA-semihypergroup. A fuzzy subset is called a fuzzy right (resp., left) hyperideal of if
(1) (resp., ),
(2) implies , for every

If is both fuzzy right hyperideal and fuzzy left hyperideal of , then is called a fuzzy hyperideal of .

Definition 10 (see [54]). Let be an ordered LA-semihypergroup. A fuzzy subset is called a fuzzy bi-hyperideal of if
(1) ,
(2) implies , for every

Definition 11 (see [54]). Let be an ordered LA-semihypergroup. A fuzzy subset is called a fuzzy interior hyperideal of if
(1) ,
(2) implies , for every

3. -Fuzzy Hyperideals

In what follows, let denote an ordered LA-semihypergroup, and unless otherwise is specified. Note that for a fuzzy point and a fuzzy subset , means that , while means that Notice that -fuzzy hyperideals are particular types of -fuzzy hyperideals.

Definition 12. A fuzzy subset of is called an -fuzzy LA-subsemihypergroup of if for all and , the following conditions hold:
(i) , ,
(ii) , for each

Definition 13. A fuzzy subset of is called an -fuzzy left hyperideal of if for all and , the following conditions hold.
(i) , ,
(ii) , for each

Definition 14. A fuzzy subset of is called an -fuzzy right hyperideal of if for all and , the following conditions hold.
(i) , ,
(ii) , for each

A fuzzy subset of is called an -fuzzy hyperideal of if it is a left hyperideal and a right hyperideal of

Definition 15. Consider a set with the following hyperoperation “” and the order “”: We give the covering relation “” and the figure of as follows: Then is an ordered LA-semihypergroup. Now let be a fuzzy subset of such that Let and Then by routine calculations it is clear that is an -fuzzy hyperideal of

Definition 16. A fuzzy subset of is called an -fuzzy interior hyperideal of if for all and , the following conditions hold:
(i) , ,
(ii) , for each

Example 17. Consider a set with the following hyperoperation “” and the order “”: We give the covering relation “” and the figure of as follows: Then is an ordered LA-semihypergroup. Now let be a fuzzy subset of such that Let and Then by routine calculations it is clear that is an -fuzzy interior hyperideal of

Definition 18. A fuzzy subset of is called an -fuzzybi-hyperideal of if for all and , the following conditions hold:
(i) , ,
(ii) , for each
(iii) , for each

A fuzzy subset of is called an -fuzzy generalizedbi-hyperideal of if it satisfies (i) and (iii) conditions of Definition 18.

Theorem 19. Let be an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of and a fuzzy subset of defined by Then
(i) is a -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of ,
(ii) is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of

Proof. (i) Consider that is an LA-subsemihypergroup of Let , and be such that Then , Since is an LA-subsemihypergroup of and , we have Thus If , then and so If , then and so Therefore Let and be such that and Then , and Since is an LA-subsemihypergroup of , we have Thus for every , If , then and so If , then and so Therefore , for all
(ii) Let , and be such that Then and Since , we have Thus If , then and so If , then and so Therefore Let and be such that and Then and Thus and , this implies that Since is an LA-subsemihypergroup of , we have Thus for every , If , then and so If , then and so Therefore , for all
The cases for left hyperideal, right hyperideal, hyperideal, interior hyperideal and bi-hyperideal can be seen in a similar way.

Corollary 20. If a nonempty subset of is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of , then the characteristic function of is a -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of

Corollary 21. A nonempty subset of is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of if and only if is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi- and hyperideal) of

If we take in Theorem 19, then we have the following corollary.

Corollary 22. Let be an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of and a fuzzy subset of defined by Then
(i) is a -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of ,
(ii) is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of

Theorem 23. Let be a fuzzy subset of . Then, is an -fuzzy LA-subsemihypergroup of if and only if
(i) , for all ;
(ii) , for all

Proof. Assume that is an -fuzzy LA-subsemihypergroup of Let be such that If , then Let and assume on the contrary that Choose such that . If , then , so , but This implies that , therefore , which is a contradiction. Hence , for all .
Suppose on the contrary that there exist such that Then there exists such that Choose such that Then and implies that and , but and So , which is a contradiction. Hence
Conversely, let for Then Now If , then and , it follows that If , then and so Thus Now, assume that for all Let and for all Then and Thus Now if , then So for every , , which implies that If , then So for every , Thus Therefore is an -fuzzy LA-subsemihypergroup of

Theorem 24. Let be a fuzzy subset of . Then, is an -fuzzy left (resp., right) hyperideal of if and only if
(i) , for all .
(ii) (resp., ), for all

Proof. proof is similar to the proof of Theorem 23.

Theorem 25. Let be a fuzzy subset of . Then, is an -fuzzy interior hyperideal of if and only if
(i) , for all .
(ii) , for all

Proof. The proof is similar to the proof of the Theorem 23.

Theorem 26. Let be a fuzzy subset of . Then, is an -fuzzy bi-hyperideal of if and only if
(i) , for all .
(ii) , for all
(ii) , for all

Proof. The is similar to the proof of Theorem 23.

A fuzzy subset of is called an -fuzzy generalizedbi-hyperideal of if it satisfies (i) and (iii) conditions of Theorem 26.

Theorem 27. A fuzzy subset of an ordered LA-semihypergroup is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of if and only if is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, and bi-hyperideal) of for all

Proof. Assume that is an -fuzzy LA-subsemihypergroup of Let be such that where Then and by Theorem 23, It follows that Let where Then and It follows from Theorem 23 that Thus for every , and so , that is Hence is an LA-subsemihypergroup of
Conversely, assume that is an LA-subsemihypergroup of for all If there exist with such that Then for some Then, but , a contradiction. Thus for all with Now, suppose that there exist such that Thus there exist such that Choose such that Then but i.e., , which is a contradiction. Hence and so is an -fuzzy LA-subsemihypergroup of

Corollary 28. A fuzzy subset of an ordered LA-semihypergroup is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of if and only if is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of for all

Example 29. Consider a set with the following hyperoperation “” and the order “”: We give the covering relation “” and the figure of as follows: Then is an ordered LA-semihypergroup. Here, , , , , and are LA-subsemihypergroups of Now let be a fuzzy subset of such that and Then by Theorem 27, is an -fuzzy LA-subsemihypergroup of for

Lemma 30. The intersection of any family of -fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, interior hyperideals, bi-hyperideals, generalized bi-hyperideals) of is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, interior hyperideal, bi-hyperideal, generalized bi-hyperideal) of

Proof. Let be a family of -fuzzy left hyperideals of and Then Since is a -fuzzy left hyperideals of Thus Hence is an -fuzzy left hyperideal of The other cases can be seen in a similar way.

Lemma 31. The union of any family of -fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, interior hyperideals, bi-hyperideals) of an ordered LA-semihypergroup is an LA-subsemihypergroup (resp., left hyperideal, right hyperideal, interior hyperideal, bi-hyperideal) of

Proof. The proof is similar to the proof of the Lemma 30.

Proposition 32. Let be an ordered LA-semihypergroup with pure left identity . Then every -fuzzy right hyperideal of is an -fuzzy left hyperideal of .

Proof. Let be an -fuzzy right hyperideal of . For , we have Thus is an -fuzzy left hyperideal of .

Proposition 33. Let be an ordered LA-semihypergroup with pure left identity . Then is an -fuzzy right hyperideal of if and only if it is an -fuzzy interior hyperideal of .

Proof. Let be an -fuzzy right hyperideal of . For , we have which implies that is an -fuzzy interior hyperideal. Conversely, for any and in we have This shows that is an -fuzzy right hyperideal of . This completes the proof.

Proposition 34. Let be an -fuzzy left hyperideal of an ordered LA-semihypergroup with pure left identity . If is an -fuzzy interior hyperideal of , then it is an -fuzzy bi-hyperideal of .

Proof. Let be an -fuzzy left hyperideal of Then for every , there exist such that Let be a pure left identity in So, This implies that Thus is an -fuzzy LA-subsemihypergroup of Now for any , Also Hence we get This shows that is an -fuzzy bi-hyperideal of

Proposition 35. Let be an ordered LA-semihypergroup with pure left identity. If is an -fuzzy subset of and is an -fuzzy left hyperideal of , then is an -fuzzy left hyperideal of

Proof. Let such that Let then Since , so implies Hence Now