Factorizable Ordered Hypergroupoids with Applications

Marty [1] proposed the notion of a hypergroup based on the multivalued operations. Hyperstructures have many applications in both pure and applied sciences. Some researchers have tried to discuss important biological phenomena in the framework of fuzzy hyperstructures. Hyperstructures were used in many disciplines such as theoretical physics, coding theory, and biology. Basic concepts and relevant applications concerning hyperstructure theory can be found in [2, 3]. Ordered semihypergroups are suggested by Heidari and Davvaz [4] and then investigated by Davvaz et al. in [5] (also, see [6–8]). ,e research about generalization of hyperideals in ordered hyperstructures is growing rapidly [9, 10]. In recent years, pseudoorders have received extensive attention in ordered hyperstructures [5, 8]. Using the notion of (weak) pseudoorder [5, 8], several examples of ordered semi (hyper) groups have been constructed in connection with ordered semihypergroups. Weak pseudoorders can remarkably support the constructions of ordered semihypergroups [8]. Breakable semihypergroups were firstly presented by Heidari and Cristea [11] in 2019. In a breakable semihypergroup, each nonempty subset is a subsemihypergroup. ,e cyclicity of the EL-hyperstructures is expressed in [12]. Recently, much attention has been paid to investigating the factorizable hyperstructures. In [13], Heidari and Cristea have suggested the concept of factorizable semihypergroups by using the concept of factorizable semigroups [14]. In this regard, Munir et al. [15] initiated the study of factorizable hypergroupoids and discussed their properties. For future work, one could extend the existing works [13, 15] to the framework of fuzzy sets. In this note, we offer basic concepts on factorizable ordered hypergroupoids. We show that if a right hyperideal R and a left hyperideal L form the factors of an ordered hypergroupoid S, then R∩L � (R◇ L]. Connection between regular and factorizable ordered semihypergroups is presented. Moreover, we prove that if an ordered hypergroupoid contains either a right (left) magnifying element, then it is factorizable. Now, we describe several information on an ordered hypergroupoid (semihypergroup) (S,◇, ≤). A hyperoperation is a mapping ◇ : S × S⟶ P∗(S), where P∗(S) denotes the family of all nonempty subsets of S. ,e couple (S,◇) is called a hypergroupoid. If ∅≠U, V⊆ S and x ∈ S, then


Introduction and Preliminaries
Marty [1] proposed the notion of a hypergroup based on the multivalued operations. Hyperstructures have many applications in both pure and applied sciences. Some researchers have tried to discuss important biological phenomena in the framework of fuzzy hyperstructures. Hyperstructures were used in many disciplines such as theoretical physics, coding theory, and biology. Basic concepts and relevant applications concerning hyperstructure theory can be found in [2,3].
Breakable semihypergroups were firstly presented by Heidari and Cristea [11] in 2019. In a breakable semihypergroup, each nonempty subset is a subsemihypergroup. e cyclicity of the EL-hyperstructures is expressed in [12]. Recently, much attention has been paid to investigating the factorizable hyperstructures. In [13], Heidari and Cristea have suggested the concept of factorizable semihypergroups by using the concept of factorizable semigroups [14]. In this regard, Munir et al. [15] initiated the study of factorizable hypergroupoids and discussed their properties. For future work, one could extend the existing works [13,15] to the framework of fuzzy sets.
In this note, we offer basic concepts on factorizable ordered hypergroupoids. We show that if a right hyperideal R and a left hyperideal L form the factors of an ordered hypergroupoid S, then R ∩ L � (R ◇ L]. Connection between regular and factorizable ordered semihypergroups is presented. Moreover, we prove that if an ordered hypergroupoid contains either a right (left) magnifying element, then it is factorizable. Now, we describe several information on an ordered hypergroupoid (semihypergroup) (S, ◇, ≤).
A hyperoperation is a mapping ◇ : S × S ⟶ P * (S), where P * (S) denotes the family of all nonempty subsets of S. e couple (S, ◇) is called a hypergroupoid. If ∅ ≠ U, V ⊆ S and x ∈ S, then A semihypergroup S is a hypergroup if the reproduction axiom is verified as (3) Definition 1 (see [4,5]). A hypergroupoid (S, ◇ ) is called an ordered hypergroupoid (semihypergroup) if Definition 2 (see [4,5] Here, a hyperideal is a left hyperideal of S being right hyperideal.

Results and Discussion
We extend the definition in [15] to the ordered case. In this section, we pay attention to the factorizable ordered hypergroupoids that is obligatory to prove our proposing results.  Table 1) and (partial) order relation ≤.
ere are four main categories within the ABO blood group system: A, B, O, and AB. e detail of blood groups is presented in [15,16], and here we will not repeat them again. An application of ordered hypergroupoid to blood groups is given.
e ABO blood groups are giving in a set Define the hyperoperation ◇ (as shown in Table 2) and (partial) order relation ≤ on S as follows.
en, (S, ◇ , ≤) is an ordered hypergroupoid. e covering relation of S is given by e Hasse diagram of S is shown in Figure 1. It is easily seen that it can be obviously seen that O is the donor blood group and AB is the recipient blood group. Also, A ≤ AB shows that people with blood group A can donate blood to people with blood groups A and AB.
Similarly, B ≤ AB shows that people with blood group B can donate blood to people with blood groups B and AB.
We need to show that R ∩ L ⊆ (R ◇ L]. Since (R, L) is a factorization of S, it follows that (R ◇ L] � S. It implies that R ∩ L ⊆ S � (R ◇ L]. □ Theorem 2. If (S, ◇ , ≤) is an ordered hypergroup, then S is regular.
Proof. Take any x ∈ S. en, So, x ≤ x ◇ a for some a ∈ S. Again, a ≤ b ◇ x for some b ∈ S. us, It implies that x ∈ (x ◇ S ◇ x]. Hence, S is regular. □ Proof. Take any x ∈ S � (U ◇ V]. en, x ≤ u ◇ v for some u ∈ U and v ∈ V. By assumption, V is regular. Consider the where U is a group and |u ◇ v| � 1 for all u ∈ U and v ∈ V. If V is an ordered hypergroup, then S is regular.
Proof. By eorem 2, V is a regular ordered semihypergroup. Now, by eorem 3, S is also regular.
where U is a group and |u ◇ v| � 1 for all u ∈ U and v ∈ V. If V is an extensive ordered semihypergroup, then S is regular.
Proof. Let V be an extensive ordered semihypergroup. en, for all t ∈ S. Hence, S is an ordered hypergroup. Now, by Corollary 1, S is regular. □ Definition 4. An element x of an ordered hypergroupoid (S, ◇ , ≤ ) is said to be a right (resp. left) magnifying element if there exists a proper subhypergroupoid U of S such that S � (U ◇ x] (resp. S � (x ◇ U]).  Table 3.
Consider an ordered hypergroup (S, ◇ , ≤) with the following (partial) order relation ≤ : Table 2: ABO blood group inheritance.   , a), (a, b), (a, c), (a, d), Note that based on the data from Table 3, Hence, d is a right (left) magnifying element of S.
An ordered hypergroupoid (S, ◇ , ≤) is called cyclic, if there exists a generator x ∈ S such that for all s ∈ S, there exists n ∈ N such that s ∈ x n . Proof. We shall prove this for right magnifying element x. We consider the following cases. Case 2. Let S be cyclic and S � 〈a〉.
As x ∈ S is a right magnifying element, S � (U ◇ x], where U is a proper subhypergroupoid of S. Since U is a proper subhypergroupoid of S, we get U � 〈a k 〉, where k ∈ N. Moreover, for l ∈ N, let x � a l . Hence, where m � k + l, which is a contradiction because (〈a m 〉] is a proper subhypergroupoid of S. Now, the proof is completed.

Conclusions
In this paper, we have described factorizable ordered hypergroupoids. We have also shown some results in this respect. An application into the fields of blood groups and factorizable ordered hypergroupoid theory was briefly introduced. We finished our study with generalized factorizable ordered hypergroupoids in hope that other factorizations such as (m, n)-factorizable ordered hypergroupoids can be discussed in the future. In the future, one can study applications of factorization in DNA coding theory.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.