On Fuzzy Ordered Hyperideals in Ordered Semihyperrings

In this paper, we introduce the concept of fuzzy ordered hyperideals of ordered semihyperrings, which is a generalization of the concept of fuzzy hyperideals of semihyperrings to ordered semihyperring theory, and we investigate its related properties.We show that every fuzzy ordered quasi-hyperideal is a fuzzy ordered bi-hyperideal, and, in a regular ordered semihyperring, fuzzy ordered quasi-hyperideal and fuzzy ordered bi-hyperideal coincide.


Introduction
The theory of algebraic hyperstructures is a well-established branch of classical algebraic theory which was initiated by Marty [1].Since then many researchers have worked on algebraic hyperstructures and developed it [2,3].A short review of this theory appears in [4][5][6][7][8].
The notion of semiring was introduced by Vandiver [9] in 1934, which is a generalization of rings.Semirings are very useful for solving problems in graph theory, automata theory, coding theory, analysis of computer programs, and so on.We refer to [10] for the information we need concerning semiring theory.In [11][12][13], quasi-ideals of semirings are studied and some properties and related results are given.In [8], Vougiouklis generalized the notion of hyperring and named it as semihyperring, where both the addition and multiplication are hyperoperations.Semihyperrings are a generalization of Krasner hyperrings.Davvaz, in [14], studies the notion of semihyperring in a general form.Ameri and Hedayati define k-hyperideals in semihyperrings in [15].In 2011, Heidari and Davvaz [16] studied a semihypergroup (, ∘) with a binary relation ≤, where ≤ is a partial order so that the monotony condition is satisfied.This structure is called an ordered semihypergroup.Properties of hyperideals in ordered semihypergroups are studied in [17].Also, the properties of fuzzy hyperideals in an ordered semihypergroup are investigated in [18,19].Yaqoop and Gulistan [20] study the concept of ordered LA-semihypergroup.In [21], Davvaz and Omidi introduce the basic notions and properties of ordered semihyperrings and prove some results in this respect.In 2018, Omidi and Davvaz [22] studied on special kinds of hyperideals in ordered semihyperrings.Some properties of hyperideals in ordered Krasner hyperrings can be found in [23].
After the introduction of fuzzy sets by Zadeh [24], reconsideration of the concept of classical mathematics began.Because of the importance of group theory in mathematics, as well as its many areas of application, the notion of fuzzy subgroup is defined by Rosenfeld [25] and its structure is investigated.This subject has been studied further by many others [26,27].Fuzzy sets and hyperstructures introduced by Zadeh and Marty, respectively, are now used in the world both on the theoretical point of view and for their many applications.There exists a rich bibliography: publications that appeared within 2015 can be found in "Fuzzy Algebraic Hyperstructures -An Introduction" by Davvaz and Cristea [28].Recently, many researchers have considered fuzzification on many algebraic structures, for example, on semigroups, rings, semirings, near-rings, ordered semigroups, semihypergroups, ordered semihypergroups, and ordered hyperrings [29][30][31][32][33][34].
Inspired by the study on ordered semihyperrings, we study the concept of fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring and we present some examples in this respect.The rest of this paper is organized as follows.

Advances in Fuzzy Systems
In the second section, we review basic concepts regarding the ordered hyperstructures.The third section is dedicated to the fuzzy ordered hyperideals and some properties.In Section 4, we introduce the concept of fuzzy ordered quasihyperideals and fuzzy ordered bi-hyperideals of an ordered semihyperring and we present some examples.We give the main theorems which characterize the ordered hyperideals, ordered quasi-hyperideals, and ordered bi-hyperideals in terms of fuzzy ordered hyperideals, fuzzy ordered quasihyperideals, and fuzzy ordered bi-hyperideals, respectively.

Terminology and Basic Properties
In what follows, we summarize some basic notions and facts about semihypergroups, semihyperrings, and ordered semihyperrings.
Let  be a nonempty set and let P * () be the set of all nonempty subsets of .A hyperoperation on  is a map ∘ :  ×  → P * () and the pair (, ∘) is called a hypergroupoid.For any  ∈  and ,  ∈ P * (), we denote A hypergroupoid (, ∘) is called a semihypergroup if for all , ,  ∈  we have ( ∘ ) ∘  =  ∘ ( ∘ ), which means that We say that a semihypergroup (, ∘) is a hypergroup if, for all  ∈ , we have  ∘  =  ∘  = .
Note that the concept of ordered semihyperring is a generalization of the concept of ordered semiring.
Now, it is easy to see that  = {0,} is an ordered bihyperideal of  but it is not an ordered quasi-hyperideal of .
The concept of a fuzzy subset of a nonempty set first was introduced by Zadeh in 1965 [36].Let  be a nonempty set.A fuzzy subset  of  is a function  :  → [0, 1].Let  and  be two fuzzy subsets of ; we say that  is contained in  and we write  ⊆ , if () ≤ () for all  ∈ , and  ∩ ,  ∪  are defined by ( ∩ )() = min{(), ()} and (∪)() = max{(), ()}.The sets , are called a level subset and strong level subset of , respectively.

On Fuzzy Ordered Hyperideals in Ordered Semihyperrings
Notice that the relationships between fuzzy sets and algebraic hyperstructures have been already considered by many researchers [18,28,30,[35][36][37][38].Recently, ordered ideals in semirings and ordered ideals in Krasner hyperrings have been already considered by Gan and Jiang [39] and Davvaz and Loeranau-Fotea [30], respectively.So, it is interesting to study fuzzy ordered hyperideals of ordered semihyperrings.
Lemma 11.Any hyperideal of an ordered semihyperring (, +, ⋅, ≤) can be realized as a level subset of some ordered fuzzy hyperideals of .
Proof.Proof is similar to Lemma 4.2 in [30].
Notice that the characteristic function of a nonempty subset  of an ordered semihyperring  is a fuzzy ordered hyperideal of  if and only if  is an ordered hyperideal of .Let  be a mapping from an ordered semihyperring  1 to an ordered semihyperring  2 .Let  be a fuzzy subset of  1 and let  be a fuzzy subset of  2 .en the inverse image  −1 () of  is a fuzzy subset of  1 defined by  −1 ()() = (()) for all  ∈  1 .e image () of  is the fuzzy subset of  2 defined by for all  ∈  2 .
Lemma 14.Let  1 and  2 be two ordered semihyperrings and let  :  1 →  2 be a strong homomorphism.
(i) If  is a fuzzy ordered hyperideal of  1 , then () is a fuzzy ordered hyperideal of  2 .
(ii) If  is a fuzzy ordered hyperideal of  2 , then  −1 () is a fuzzy ordered hyperideal of  1 .
Proof.It is straightforward.

Fuzzy Ordered Bi-Hyperideals and Fuzzy Ordered Quasi-Hyperideals of Ordered Semihyperrings
In this section, we define the concepts of fuzzy ordered bihyperideal and fuzzy ordered quasi-hyperideal in ordered semihyperrings and give relationships between them.Let (, +, ⋅, ≤) be an ordered semihyperring and  ∈ .We denote [30].For fuzzy subsets  and  of a semihyperring , we define the fuzzy subset  ∘  of  by letting  ∈ ; We denote the constant function 1 :  → [0, 1] defined by 1() = 1 for all  ∈  [30].

Theorem 16. A fuzzy subset 𝜇 of an ordered semihyperring 𝑅 is a fuzzy ordered bi-hyperideal of 𝑅 if and only if the set 𝜇
Proof.The proof is similar to the proof of Theorem 12.
Definition .Let (, +, ⋅, ≤) be an ordered semihyperring and let  be a fuzzy subset of .Then,  is called a fuzzy ordered quasi-hyperideal of  if the following conditions hold: The following theorem can be proved in a similar way in the proof of Theorem 4.8 of [30].Let  be a fuzzy ordered hyperideal of  and  ∈ .We have If   = 0, then it is easy to see that min {( ∘ 1) () , (1 ∘ ) ()} ⊆  () .
If   ̸ = 0, then there exist ,  ∈  such that  ≤  ⋅ .Then there exists  ∈ ⋅ such that  ≤ .Since  is a fuzzy ordered of , we have That is, () ≥ min{(), ()}.On the other hand, That is, the condition (ii) of Definition 17 is satisfied.Thus  is a fuzzy ordered quasi-hyperideal of .
The following example shows that the converse of Theorem 20 is not true in general.

(25)
We have Since  and {0, } are ordered bi-hyperideal of , then   is an ordered bi-hyperideal of  for all  ∈ [0, 1].Hence  is a fuzzy ordered bi-hyperideal of  by Theorem 16.But it is not a fuzzy ordered quasi-hyperideal of .

Conclusion
In the structural theory of fuzzy algebraic systems, fuzzy ideals with special properties always play an important role.In this paper, we study fuzzy ordered hyperideals, fuzzy ordered quasi-hyperideals, and fuzzy ordered bi-hyperideals of an ordered semihyperring.We characterize regular ordered semihyperrings by the properties of these fuzzy hyperideals.As a further work, we will also concentrate on characterizations of different classes of ordered semihyperrings in terms of fuzzy interior hyperideals.
(i) Every fuzzy ordered hyperideal of  is a fuzzy ordered quasi-hyperideal of  (ii) Every fuzzy ordered quasi-hyperideal of  is a fuzzy ordered bi-hyperideal of  Proof.(i) Only we show that the condition (ii) of Definition 17 is satisfied.
An ordered semihyperring (, +, ⋅, ≤) is called regular, if, for every  ∈ , there exists  ∈  such that  ≤  ⋅  ⋅ .Let (, +, ⋅, ≤) be a regular ordered semihyperring and let  be a fuzzy subset of .en,  is a fuzzy ordered quasi-hyperideal of  if and only if  is a fuzzy ordered bihyperideal of .Proof."⇒" Assume that  is a fuzzy ordered quasihyperideal of .It is clear that  is a fuzzy ordered bihyperideal of  by Theorem 20 (ii).