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

Research Article

On -Fuzzy Hyperideals in Ordered LA-Semihypergroups

Muhammad Azhar,¹ Naveed Yaqoob,² Muhammad Gulistan,¹ and Mohammed M. Khalaf³

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; nayaqoob@ymail.com

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. [37–40], 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

Thus Let Then

If , then Now for each , there exist such that Since is an -fuzzy left hyperideal of , , that is Thus

So Thus is an -fuzzy left hyperideal of

Definition 36. Let and be any two fuzzy subsets of . We define the product by

Definition 37. The symbols , and mean the following fuzzy subsets of :

Denote by the family of all fuzzy subsets in

Theorem 38. Let be an ordered LA-semihypergroup. Then the set is an ordered LA-semihypergroup.

Proof. Clearly is closed. Let , and be in and let be any element of such that it is not expressible as product of two elements in . Then we have

Let . Then there exist and in such that Therefore by using left invertive law, we have

Hence is an ordered LA-semihypergroup. Assume that and let for any ; then

Similarly we can show that Let . Then there exist and in such that , therefore

Similarly we can show that It is easy to see that is a poset. Thus is an ordered LA-semihypergroup.

Theorem 39. Let be an ordered LA-semihypergroup. Then the property

holds in , for all and in

Proof. proof is straightforward.

Theorem 40. If an ordered LA-semihypergroup has a pure left identity, then the following properties hold in
(i) ,
(ii) ,
for all and in

Proof. The is straightforward.

Proposition 41. Let be an -fuzzy right hyperideal of and is an -fuzzy left hyperideal of . Then .

Proof. Let be an -fuzzy right hyperideal of and an -fuzzy left hyperideal of . Let and suppose that there exist such that . Then

Let us suppose that there do not exist any such that . Then, Hence .

Lemma 42. Let be an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal, generalized bi-hyperideal) of Then is a fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal, generalized bi-hyperideal) of

Proof. Let be an -fuzzy right hyperideal of Then for all , we have and . This implies that

So Also

So Thus is a fuzzy right hyperideal of Other cases can be seen in a similar way.

For an ordered LA-semihypergroups , the fuzzy subset is defined as follows:

Proposition 43. For a fuzzy subset of , the following conditions are true for all
(i) If is a fuzzy LA-subsemihypergroup of , then and ,
(ii) If is a fuzzy left hyperideal of , then and ,
(iii) If is a fuzzy right hyperideal of , then and ,
(iv) If is a fuzzy interior hyperideal of , then and ,
(v) If is a fuzzy generalized bi-hyperideal of , then and ,
(vi) If is a fuzzy bi-hyperideal of , then , and

Proof. (i) Let be a fuzzy LA-subsemihypergroup of If , then

If , then

Thus For , we have This implies The proofs of (ii) to (vi) can be seen in a similar way.

Proposition 44. Let be an ordered LA-semihypergroup with pure left identity. Then for any -fuzzy left hyperideal , which is idempotent in , the following properties hold:
(i) is an -fuzzy interior hyperideal;
(ii) is an -fuzzy bi-hyperideal.

Proof. Let be an -fuzzy left hyperideal of and
(i) We have

This implies that is an -fuzzy interior hyperideal
(ii) We have

This implies that is an -fuzzy bi-hyperideal of

Theorem 45. Let be a regular ordered LA-semihypergroup. Then for every -fuzzy generalized bi-hyperideal of .

Proof. Let us suppose that is a regular ordered LA-semihypergroup and let be an -fuzzy generalized bi-hyperideal of . Let and so for some . Thus we have

Therefore, . Since , by Proposition 43(v), we have for every -fuzzy generalized bi-hyperideal of .

Proposition 46. If is an -fuzzy right hyperideal of a regular ordered LA-semihypergroup with pure left identity , then holds for all .

Proof. Let be an -fuzzy right hyperideal of a regular ordered LA-semihypergroup with pure left identity . Let Since is regular, and for some . Now by using the medial and paramedial laws, we get

Since is an -fuzzy right hyperideal, for every , we have

Again by using the medial and paramedial laws, we get

Since is an -fuzzy right hyperideal, for every , we have

This shows that holds for all .

Theorem 47. Let be a regular ordered LA-semihypergroup with pure left identity . Then is an -fuzzy left hyperideal of if and only if is an -fuzzy bi-hyperideal of

Proof. Let be an -fuzzy left hyperideal of . Let . We have

This shows that is an -fuzzy generalized bi-hyperideal of , and clearly is an -fuzzy LA-subsemihypergroup. Therefore is an -fuzzy bi-hyperideal of
Conversely, assume that is an -fuzzy bi-hyperideal of . Let Since is regular, by the left invertive law and medial law, we have

Thus for every , we have

Hence is an -fuzzy left hyperideal of

4. Homomorphism

Let and be two ordered LA-semihypergroups and a mapping from into . is called isotone if , implies . is said to be inverse isotone if , implies [each inverse isotone mapping is 1-1]. is called a homomorphism if it is isotone and satisfies , for all Moreover, is said to be isomorphism if it is onto homomorphism and inverse isotone.

Definition 48. Let and be two ordered LA-semihypergroups and a mapping from into Let and be fuzzy subsets of and , respectively. Then the image of is defined by

And the inverse image of is defined by

Theorem 49. Let and be two ordered LA-semihypergroups and a mapping from onto Let and be -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of and , respectively. Then
(i) is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of , provided is inverse isotone.
(ii) is an -fuzzy LA-subsemihypergroup (resp., left hyperideal, right hyperideal, hyperideal, interior hyperideal, bi-hyperideal) of
(iii) The mapping defines a one-to-one correspondence between the set of all -fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, hyperideals, interior hyperideals, bi-hyperideals) of and the set of all -fuzzy LA-subsemihypergroups (resp., left hyperideals, right hyperideals, hyperideals, interior hyperideals, bi-hyperideals) of , provided is inverse isotone.

Proof. The proof is ward.

Theorem 50. Let be a surjective homomorphism from an ordered LA-semihypergroup to an ordered LA-semihypergroup . If contains a pure left identity , then
(i) the image of -fuzzy interior hyperideal of is an -fuzzy right hyperideal of , provided is inverse isotone.
(ii) the preimage of an -fuzzy right hyperideal of is an -fuzzy interior hyperideal of

Proof. (i) Let be an -fuzzy interior hyperideal of and let . Then there exist such that , and Now, we have

Let Since is inverse isotone, there exist unique such that , and Thus we have

This shows that the image of an -fuzzy interior hyperideal of is an -fuzzy right hyperideal of The proof of (ii) can be seen in a similar way.

In the following example we show that in an ordered LA-semihypergroup without pure left identity, the image of an -fuzzy subset under can be or cannot be an -fuzzy right hyperideal of .

Example 51. Let and be two ordered LA-semihypergroups defined by the following hyperoperations “” and the orders “”:

We define a homomorphism by

We take a fuzzy subset of as , , and By routine calculation one can see that the image of under is an -fuzzy right hyperideal of for , but is not an -fuzzy interior hyperideal of

Theorem 52. Let be a homomorphism from an ordered LA-semihypergroup to an ordered LA-semihypergroup . If contains a pure left identity , then the preimage of every -fuzzy interior hyperideal of is an -fuzzy generalized bi-hyperideal of , provided is inverse isotone.

Proof. The proof is similar to the proof of Theorem 50(i).

5. Conclusions

Fuzzification of algebraic hyperstructures plays an important role in mathematics with wide range of applications in many disciplines such as computer sciences, chemistry, engineering, and medical diagnosis. In this paper, we have introduced the concept of -fuzzy hyperideals in ordered LA-semihypergroup and investigated their related properties. We hope that this work would offer foundation for further study of the theory on ordered LA-semihypergroup and fuzzy ordered LA-semihypergroup. The obtained results probably can be applied in various fields such as computer sciences, control engineering, coding theory, theoretical physics, and chemistry. In our future research, we will consider the characterization of intraregular ordered LA-semihypergroup in terms of -fuzzy hyperideals.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

References

F. Marty, “Sur une generalization de la notion de group,” in Proceedings of the 8th Congress of Mathematics, pp. 45–49, Stockholm, Sweden, 1934.

P. Corsini, Prolegomena of Hypergroup Theory, Aviain editore, Tricesimo, Italy, 2nd edition, 1993.

T. Vougiouklis, Hyperstructures and Their Representations, vol. 115, Hadronic Press, Palm Harbor, Fla, USA, 1994.

P. Bonansinga and P. Corsini, “On semihypergroup and hypergroup homomorphisms,” Bollettino dell Unione Matematica Italiana. Sezione B, vol. 1, no. 2, pp. 717–727, 1982.

P. Corsini, “Sur les semi-hypergroupes,” Atti della Società Peloritana di Scienze Fisiche Matematiche e Naturali, vol. 26, no. 4, pp. 363–372, 1980 (French).

B. Davvaz, “Some results on congruences on semihypergroups,” Bulletin of the Malaysian Mathematical Sciences Society. Second Series, vol. 23, no. 1, pp. 53–58, 2000.

B. Davvaz, W. A. Dudek, and S. Mirvakili, “Neutral elements, fundamental relations and n-ary hypersemigroups,” International Journal of Algebra and Computation, vol. 19, no. 4, pp. 567–583, 2009.

B. Davvaz and N. S. Poursalavati, “Semihypergroups and S-hypersystems,” Pure Mathematics and Applications, vol. 11, no. 1, pp. 43–49, 2000.

C. Gutan, “Simplifiable semihypergroups,” in Algebraic Hyperstructures and Applications (Xanthi, 1990), pp. 103–111, World Scientific Publishing, Teaneck, NJ, USA, 1991.

A. Hasankhani, “Ideals in a semihypergroup and Green’s relations,” Ratio Mathematica, no. 13, pp. 29–36 (2000), 1999.

K. Hila, B. Davvaz, and K. Naka, “On quasi-hyperideals in semihypergroups,” Communications in Algebra, vol. 39, no. 11, pp. 4183–4194, 2011.

S. V. Onipchuk, “Regular semihypergroups,” Russian Academy of Sciences. Sbornik Mathematics, vol. 183, no. 6, pp. 43–54, 1992.

K. Hila and J. Dine, “On hyperideals in left almost semihypergroups,” ISRN Algebra, Article ID 953124, pp. 1–8, 2011.

N. Yaqoob, P. Corsini, and F. Yousafzai, “On intra-regular left almost semihypergroups with pure left identity,” Journal of Mathematics, vol. 2013, Article ID 510790, pp. 1–10, 2013.

F. Yousafzai and P. Corsini, “Some charactrization problems in LA-semihypergroups,” Journal of Algebra, Number Theory: Advances and Applications, vol. 10, no. 12, pp. 41–55, 2013.

F. Yousafzai, K. Hila, P. Corsini, and A. Zeb, “Existence of non-associative algebraic hyper-structures and related problems,” Afrika Matematika, vol. 26, no. 5-6, pp. 981–995, 2015.

D. Heidari and B. Davvaz, “On ordered hyperstructures,” Politehnica University of Bucharest. Scientific Bulletin. Series A. Applied Mathematics and Physics, vol. 73, no. 2, pp. 85–96, 2011.

M. Bakhshi and R. A. Borzooei, “Ordered polygroups,” Ratio Mathematica, vol. 24, pp. 31–40, 2013.

J. Chvalina, “Commutative hypergroups in the sense of Marty and ordered sets,” in Proceedings of the Summer School on General Algebra and Ordered Sets, pp. 19–30, Olomouc (Czech Republic), 1994.

S. Hoskova, “Upper order hypergroups as a reflective subcategory of subquasi-order hypergroups,” Italian Journal of Pure and Applied Mathematics, no. 20, pp. 215–222, 2006.

M. Kondo and N. Lekkoksung, “On intra-regular ordered Γ-semihypergroups,” International Journal of Mathematical Analysis, vol. 7, no. 25-28, pp. 1379–1386, 2013.

M. Novak, “EL-hyperstructures,” Ratio Mathematica, vol. 23, pp. 65–80, 2012.

N. Yaqoob and M. Gulistan, “Partially ordered left almost semihypergroups,” Journal of the Egyptian Mathematical Society, vol. 23, no. 2, pp. 231–235, 2015.

L. A. Zadeh, “Fuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965.

R. Ameri, “Fuzzy hypervector spaces over valued fields,” Iranian Journal of Fuzzy Systems, vol. 2, no. 1, pp. 37–47, 80, 2005.

R. Ameri, H. Hedayati, and A. Molaee, “On fuzzy hyperideals of Γ-hyperrings,” Iranian Journal of Fuzzy Systems, vol. 6, no. 2, pp. 47–59, 2009.

R. Ameri and H. Hedayati, “Fuzzy isomorphism and quotient of fuzzy subpolygroups,” Quasigroups and Related Systems, vol. 13, no. 2, pp. 175–184, 2005.

P. Corsini, M. Shabir, and T. Mahmood, “Semisimple semihypergroups in terms of hyperideals and fuzzy hyperideals,” Iranian Journal of Fuzzy Systems, vol. 8, no. 1, pp. 95–111, 168, 2011.

I. Cristea, “Hyperstructures and fuzzy sets endowed with two membership functions,” Fuzzy Sets and Systems, vol. 160, no. 8, pp. 1114–1124, 2009.

B. Davvaz, “Fuzzy hyperideals in semihypergroups,” Italian Journal of Pure and Applied Mathematics, no. 8, pp. 67–74, 2000.

B. Davvaz, “Extensions of fuzzy hyperideals in H V-semigroups,” Soft Computing, vol. 11, no. 9, pp. 829–837, 2007.

B. Davvaz and V. Leoreanu-Fotea, “Structures of fuzzy Γ-hyperideals of Γ-semihypergroups,” Journal of Multiple-Valued Logic and Soft Computing, vol. 19, no. 5-6, pp. 519–535, 2012.

A. Khan, M. Farooq, M. Izhar, and B. Davvaz, “Fuzzy hyperideals of left almost semihypergroups,” International Journal of Analysis and Applications, vol. 15, no. 2, pp. 155–171, 2017.

J. Zhan, “On properties of fuzzy hyperideals in hypernear-rings with t-norms,” Journal of Applied Mathematics & Computing, vol. 20, no. 1-2, pp. 255–277, 2006.

B. Pibaljommee, K. Wannatong, and B. Davvaz, “An investigation on fuzzy hyperideals of ordered semihypergroups,” Quasigroups and Related Systems, vol. 23, no. 2, pp. 297–308, 2015.

B. Pibaljommee and B. Davvaz, “Characterizations of (fuzzy) bi-hyperideals in ordered semihypergroups,” Journal of Intelligent & Fuzzy Systems, vol. 28, no. 5, pp. 2141–2148, 2015.

J. Tang, A. Khan, and Y. Luo, “Characterizations of semisimple ordered semihypergroups in terms of fuzzy hyperideals,” Journal of Intelligent & Fuzzy Systems, vol. 30, no. 3, pp. 1735–1753, 2016.

J. Tang, B. Davvaz, and X. Xie, “A study on (fuzzy) quasi-γ-hyperideals in ordered γ-semihypergroups,” Journal of Intelligent & Fuzzy Systems, vol. 32, no. 6, pp. 3821–3838, 2017.

J. Tang, B. Davvaz, and Y. Luo, “A study on fuzzy interior hyperideals in ordered semihypergroups,” Italian Journal of Pure and Applied Mathematics, no. 36, pp. 125–146, 2016.

J. Tang, B. Davvaz, and Y. Luo, “Hyperfilters and fuzzy hyperfilters of ordered semihypergroups,” Journal of Intelligent & Fuzzy Systems, vol. 29, no. 1, pp. 75–84, 2015.

N. Tipachot and B. Pibaljommee, “Fuzzy interior hyperideals in ordered semihypergroups,” Italian Journal of Pure and Applied Mathematics, no. 36, pp. 859–870, 2016.

V. Murali, “Fuzzy points of equivalent fuzzy subsets,” Information Sciences, vol. 158, pp. 277–288, 2004.

P. M. Pu and Y. M. Liu, “Fuzzy topology. I. Neighborhood structure of a fuzzy point and Moore-Smith convergence,” Journal of Mathematical Analysis and Applications, vol. 76, no. 2, pp. 571–599, 1980.

S. K. Bhakat and P. Das, “On the definition of a fuzzy subgroup,” Fuzzy Sets and Systems, vol. 51, no. 2, pp. 235–241, 1992.

S. K. Bhakat and P. Das, “(∈,∈∨q)-fuzzy subgroup,” Fuzzy Sets and Systems, vol. 80, no. 3, pp. 359–368, 1996.

Y. B. Jun, A. Khan, and M. Shabir, “Ordered semigroups characterized by their (∈,∈∨q)-fuzzy bi-ideals,” Bulletin of the Malaysian Mathematical Sciences Society. Second Series, vol. 32, no. 3, pp. 391–408, 2009.

M. Shabir, Y. B. Jun, and Y. Nawaz, “Semigroups characterized by (∈,∈∨qk)-fuzzy ideals,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1473–1493, 2010.

S. Azizpour and Y. Talebi, “Characterizations of interior hyperideals of semihypergroups towards fuzzy points,” in Proceedings of the 46th Annual Iranian Mathematics Conference, pp. 1–4, 2015.

S. Azizpour and Y. Talebi, “Some results on union of (∈,∈∨qk)-fuzzy interior hyperideals of semihypergroups,” in Proceedings of the 4th Iranian Joint Congress on Fuzzy and Intelligent Systems, CFIS 2015, pp. 1–4, September 2015.

X. Huang, Y. Yin, and J. Zhan, “Characterizations of semihyperrings by their (∈γ,∈γ,∨qδ)-fuzzy hyperideals,” Journal of Applied Mathematics, Art. ID 348203, 13 pages, 2013.

M. Shabir and T. Mahmood, “Semihypergroups characterized by (∈,∈∨qk)-fuzzy hyperideals,” Journal of Intelligent & Fuzzy Systems, vol. 28, no. 6, pp. 2667–2678, 2015.

M. Shabir and T. Mahmood, “Semihypergroups characterized by (∈γ,∈γ,∨qδ)-fuzzy hyperideals,” Journal of Intelligent & Fuzzy Systems, vol. 28, no. 6, pp. 2667–2678, 2015.

M. Azhar, M. Gulistan, N. Yaqoob, and S. Kadry, “On fuzzy ordered LA-semihypergroups,” International Journal of Analysis and Applications, vol. 16, no. 2, pp. 276–289, 2018.

M. Azhar and M. Gulistan, “On fuzzy hyperideals in ordered LA-semihypergroups,” Discrete Dynamics in Nature and Society, In press.