The L-ordered L-semihypergroups

Abstract This study pursues an investigation on L-semihypergroups equipped with an L-order. First, the concept of L-ordered L-semihypergroups is introduced by L-posets and L-semihypergroups, and some related results are obtained. Then, prime, weakly prime, and semiprime L-hyperideals of L-ordered L-semihypergroups are studied. Moreover, the relationships among the three types of L-hyperideals are established. Finally, the intra-regular L-ordered L-semihypergroups are characterized in terms of these L-hyperideals. The results of the study show that some well-known results on ordered semihypergroups also hold in the case of L-ordered L-semihypergroups.


Introduction
The theory of ordered semihypergroups generated by the fusion of ordered structures and algebraic hyperstructures is a newly developed research field of ordered algebra theory. In fact, it is a generalization of the theory of ordered semigroups. More precisely, an ordered semihypergroup [1] is a semihypergroup ( ∘) S, together with an order ≤ that is compatible with the hyperoperation, meaning that for all ∈ x y z S , , , ≤ ⇒ ∘ ≤ ∘ ∘ ≤ ∘ x y x z y z z x z y and , where ∘ ≤ ∘ x z y z means for all ∈ ∘ u x z there is ∈ ∘ v y z such that ≤ u v and ∘ ≤ ∘ z x z y is defined similarly. Later on, a lot of researchers focused on this topic (see [2][3][4][5][6][7][8][9][10][11][12]), for instance, Changphas and Davvaz [2] investigated the properties of hyperideals in an ordered semihypergroup; Davvaz et al. [3] studied the relationship between ordered semihypergroups and ordered semigroups by pseudoorders; Kehayopulu [4] introduced prime, weakly prime, and semiprime hyperideals of ordered semihypergroups and described ordered semihypergroups with these hyperideals. This theory can be penetrated into hyperalgebraic theories such as ordered hypergroups, ordered hyperrings, and ordered semihyperrings. It also plays a positive role in promoting the development of other disciplines.
Once the concept of fuzzy sets was put forward, it was quickly penetrated into other branches of mathematics and was widely used in many fields [13][14][15]. In computer science, an order usually expresses qualitative information between elements, so it cannot provide quantitative information needed for actual calculation. The introduction of fuzzy orders [16] makes up for this deficiency. It has been widely used in decision making, intelligent control, and so on. In particular, the combination of fuzzy orders and algebraic systems has appeared in recent years. For example, Šešlja et al. [17] induced fuzzy orders on fuzzy subsets through the classical order on a sequence group and proposed a concept of fuzzy ordered groups. The "fuzzy" is currently employed in the theory, which builds over the unit interval [ ] 0, 1 . If it builds over a complete lattice L, then the fuzzy order is usually called an L-order. In the last decade, many authors investigated L-order and applied it to various branches of mathematics and computer sciences (see [18][19][20][21][22][23][24][25]), such as its application in the semigroup theory. In 2012, Hao [21] and Wang [22] applied the theory of L-orders to ordered semigroups to construct the L-ordered semigroups, respectively. In 2015, Borzooei et al. [23] defined L-ordered groups and L-lattice ordered groups by directly combining the concept of L-orders with groups. All the aforementioned investigations focus mainly on L-order structures, so Huang et al. [24] studied the L-ordered semigroups from the perspective of more emphasis on algebras in 2018.
On the other hand, because the combination of fuzzy mathematics and hyperalgebra fully reflects the uncertainty of objective things themselves, fuzzy hyperalgebras have been concerned by many scholars since it was created. At present, it is one of the methods to study fuzzy hyperalgebras by defining and studying fuzzy hyperoperations. Similar to crisp hyperoperations, fuzzy hyperoperations map a pair of elements on a nonempty set S to a fuzzy subset of S. This method was first proposed by Corsini and Tofan [25]. Based on this method, Sen et al. [26] introduced the fuzzy semihypergroups; Leoreanu-Fotea and Davvaz [27] studied fuzzy hyperrings; Yin et al. [28] studied L-fuzzy hypergroups and L-fuzzy supermodules, and so on.
Motivated by the works of the L-ordered semigroups and fuzzy semihypergroups, we attempt in the present study to study the combination of L-orders and L-semihypergroups in detail.
The contents are arranged as follows: in Section 2, some basic notions and conclusions that will be used throughout this study are listed. In Section 3, the concept of an L-ordered L-semihypergroup is proposed. In addition, the relationship between ordered semihypergroups and L-ordered L-semihypergroups is discussed. In Section 4, the notions of L-left (resp. L-right, L-) hyperideals of an L-ordered L-semihypergroup are introduced and investigated. In Section 5, prime, weakly prime, and semiprime L-hyperideals of L-ordered L-semihypergroups are introduced and studied. It is proved that: (1) the L-hyperideals of an L-ordered L-semihypergroup are weakly prime if and only if they are idempotent and form a weak chain; (2) the L-hyperideals of an L-ordered L-semihypergroup are prime if and only if they form a weak chain and the L-ordered L-semihypergroup is intra-regular. Finally, some conclusions are presented in Section 6.

Preliminaries
For the convenience of the reader, in this section, some basic concepts are reviewed. Because a complete residuated lattice is significant in fuzzy logic, it is used as the structure of truth values throughout this study. If no other conditions are imposed, in the sequel, L always denotes a complete residuated lattice.
is called a nonempty L-subset of S and the symbol L S⁎ denotes the set of all nonempty L-subsets of S. For ∈ α L and ⊆ A L, the L-subset α A is defined by In particular, when = α 1, α A is said to be the characteristic function of A, denoted by χ A and when = { } A x , α A is said to be an L-point with support x and value α and denoted by is a mapping, called an L-order, that satisfies for any ∈ x y z S , , , For a given L-poset ( ) S R , , define a binary mapping L S is defined as follows: x S It is simple to check that Definition 2.4. [26] An L-semihypergroup is a pair ( ∘) S, such that S is a nonempty set and ∘ × → S S L : S⁎ is a mapping, called an L-hyperoperation and written as ( , that satisfies for any ∈ x y z S , , , for all ∈ μ ν L , S⁎ and all ∈ x S.
0, 1 and L S⁎ is replaced by L S , the L-semihypergroup is the fuzzy semihypergroup defined in [26].

L-ordered L-semihypergroups
In this section, we introduce the concept of the L-ordered L-semihypergroups and give some examples to explain it. Moreover, we also discuss the relationship between L-ordered L-semihypergroups and ordered semihypergroups.
1. An L-ordered L-semihypergroup is a triple ( ∘ ) S R , , consisting of a nonempty set S together with an L-relation R and an L-hyperoperation ∘ on S such that is an L-ordered semigroup, which is introduced in [24]. Conversely, for every L-ordered semigroup ( ) S R , •, , we can define an L-hyperoperation on S as follows: for all ∈ x y z S , , , • .

  
Then, for all ∈ x y z S , , , is an L-ordered L-semihypergroup. for all ∈ x y S , . Then, ( ∘) S, is a L-semihypergroup by [5,6]. Moreover, for all ∈ x y z S , , , we have is an L-ordered L-semihypergroup.
is an L-ordered L-semihypergroup.
is an L-ordered L-semihypergroup.
Starting now with an L-ordered L-semihypergroup ( ∘ ) S R , , and defining the following hyperoperation and the order on S: for all ∈ x y S , , 1, then we can obtain an ordered semihypergroup, as follows.
is an L-ordered L-semihypergroup, then ( ≤ ) S, •, R is an ordered semihypergroup, which is called the associated ordered semihypergroup.
Combining the aforementioned arguments, we have that ( ≤ ) S, •, R is an ordered semihypergroup. □ Starting now with an ordered semihypergroup ( ≤) S, •, and defining the following L-hyperoperation and the L-order on S: for all ∈ x y S , , ∘ = x y χ x y • and then we can obtain an L-ordered L-semihypergroup, as follows.
is an ordered semihypergroup, then ( ∘ ) is an L-ordered L-semihypergroup, which is called the associated L-ordered L-semihypergroup.
(2) We can easily check that , .
This completes the proof. □

L-hyperideals of L-ordered L-semihypergroups
In this section, we introduce the concept of L-left (resp. L-right, L-) hyperideals of an L-ordered L-semihypergroup and develop some characterizations for them, some of which are useful in sequel.
be an L-ordered L-semihypergroup and ∈ μ L S⁎ . μ is called an L-left (resp. Clearly, Definition 4.1 is a generalization for the concept of left (resp. right) hyperideals on crisp ordered semihypergroups. It should be noticed that, whenever a statement is made about L-left hyperideals, it is to be understood that the analogous statement is also made about L-right hyperideals. By an L-hyperideal, we mean the one which is both an L-left and L-right hyperideal.
By [26], we have that an L-lower subset μ is an L-left (resp. L-right) hyperideal if and only if ∘ ⊆ x μ μ (resp. ∘ ⊆ μ x μ) for all ∈ x S. Moreover, it is easily checked that χ S is an L-hyperideal. By [26] and the properties of L-lower subsets, we have the following results.
The L-ordered L-semihypergroups  557 be an L-ordered L-semihypergroup. Then, (2) for all L-hyperideals μ and ν, ( ∘ ] μ ν is also an L-hyperideal; Proof. (1) Let ∈ x S. Then, we have be an L-ordered L-semihypergroup and μ a nonempty L-subset of S. Then, S is the L-hyperideal generated by μ. be an L-ordered L-semihypergroup. If μ is an L-left hyperideal and ν is an L-right hyperideal of S, then ( ∘ ] μ ν is an L-hyperideal of S.
Proof. Since μ and ν are nonempty L-subsets of S, , which implies that ∘ μ ν is nonempty. Thus, ( ∘ ] μ ν is also nonempty. In addition, Proof. First, we show that ⋃ ∈ μ i I i is an L-hyperideal. In fact, it is obvious that ⋃ ∈ μ i I i is nonempty by Second, we show that ⋂ ∈ μ i I i is an L-hyperideal. Again, because every μ i is an L-hyperideal, for all ∈ x y X , ,
be an L-ordered L-semihypergroup and ∈ δ L S⁎ . for all ∈ x y S , and all ∈ α β L , ; (2) δ is semiprime if and only if Proof.
Proof. Let δ be an prime L-hyperideal, then it is obvious that δ is weakly prime and semiprime.
In particular, suppose that ( ∘ ) S R , , is commutable. Then, every weakly prime L-hyperideal is prime. In fact, let δ be a weakly prime L-hyperideal and μ ν , L-hyperideals, then  Finally, we defined and investigated L-hyperideals. In particular, we studied prime, weakly prime, and semiprime L-hyperideals of L-ordered L-semihypergroups and established the relationships among them. Following this study, we will investigate the concepts of L-order-congruences and L-pseudoorders on an L-ordered L-semihypergroup and give out homomorphism theorems of L-ordered L-semihypergroups by L-pseudoorders.