Abstract
In this study, we propose the concept of factorizable ordered hypergroupoids (semihypergroups) and present several of its properties. Our goal is to construct ordered hypergroupoid from blood groups together with some other information. Finally, we discuss right magnifying elements for further research.
1. 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].
Ordered semihypergroups are suggested by Heidari and Davvaz [4] and then investigated by Davvaz et al. in [5] (also, see [6–8]). The 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. The cyclicity of the -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 and a left hyperideal form the factors of an ordered hypergroupoid , then . 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) .
A hyperoperation is a mapping , where denotes the family of all nonempty subsets of . The couple is called a hypergroupoid. If and , then
A semihypergroup is a hypergroupoid , where
A semihypergroup is a hypergroup if the reproduction axiom is verified as
A hyperoperation on is extensive [12] if for all .
Definition 1 (see [4, 5]). A hypergroupoid is called an ordered hypergroupoid (semihypergroup) if(1) is an ordered set(2) implies and , for all If , then
Here, of an ordered hypergroupoid (semihypergroup) is called a subhypergroupoid (subsemihypergroup) of if for every and is the relation restricted to . Set
If , then(1)(2)If , then (3)
An ordered hypergroupoid is regular if for every .
Definition 2 (see [4, 5]). A set of an ordered hypergroupoid is left (resp. right) hyperideal if(1) (resp. )(2)Here, a hyperideal is a left hyperideal of being right hyperideal.
2. 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.
Definition 3. An ordered hypergroupoid (semihypergroup) is said to be factorizable if there exist two different proper subhypergroupoids (subsemihypergroups) and of such that . The pair is called a factorization of , with factors and .
Example 1. Consider an ordered hypergroup with the following hyperoperation (as shown in Table 1) and (partial) order relation .The subhypergroups and then form the factorization of as
There are four main categories within the ABO blood group system: , and . The 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.
Example 2. The ABO blood groups are giving in a setDefine the hyperoperation (as shown in Table 2) and (partial) order relation on as follows.Then, is an ordered hypergroupoid. The covering relation of is given byThe Hasse diagram of is shown in Figure 1.
It is easily seen that(1)Clearly, and . From the figure, it can be obviously seen that is the donor blood group and is the recipient blood group. Also, shows that people with blood group can donate blood to people with blood groups and . Similarly, shows that people with blood group can donate blood to people with blood groups and .(2)The subhypergroupoids and form the factorization of as .(3) is a simple ordered hypergroupoid. It can be seen that is simple if and only if for every .
Theorem 1. Let be an ordered hypergroupoid. If the right hyperideal and the left hyperideal form the factors of , then
Proof. As and , we have . Now, we obtainWe need to show that . Since is a factorization of , it follows that . It implies that .
Theorem 2. If is an ordered hypergroup, then is regular.
Proof. Take any . Then,So, for some . Again, for some . Thus,It implies that . Hence, is regular.
Theorem 3. Let be an ordered semihypergroup, where . Let be factorizable as , where is a group and for all and . If is a regular ordered semihypergroup, then is also a regular ordered semihypergroup.
Proof. Take any . Then, for some and . By assumption, is regular. Consider the element and assume that for some . So,Thus, for some . Hence, is regular.
Corollary 1. Let be an ordered semihypergroup, where . Let be factorizable as , where is a group and for all and . If is an ordered hypergroup, then is regular.
Proof. By Theorem 2, is a regular ordered semihypergroup. Now, by Theorem 3, is also regular.
An ordered hypergroupoid is said to be extensive if for all .
Corollary 2. Let be an ordered semihypergroup, where . Let be factorizable as , where is a group and for all and . If is an extensive ordered semihypergroup, then is regular.
Proof. Let be an extensive ordered semihypergroup. Then, for all . So,for all . Hence, is an ordered hypergroup. Now, by Corollary 1, is regular.
Definition 4. An element of an ordered hypergroupoid is said to be a right (resp. left) magnifying element if there exists a proper subhypergroupoid of such that (resp. ).
Example 3. In Example 2, is a subhypergroupoid of . We haveThus, and are right magnifying elements of . We consider next the subhypergroupoid . Observe that if , then and are right magnifying elements of .
Example 4. Given a hypergroup , let the hyperoperation be defined in Table 3.
Consider an ordered hypergroup with the following (partial) order relation :Note that based on the data from Table 3, is a subhypergroup of . If , thenHence, is a right (left) magnifying element of .
An ordered hypergroupoid is called cyclic, if there exists a generator such that for all , there exists such that .
Proposition 1. Let be an ordered hypergroupoid. If contains either a right (left) magnifying element , then is factorizable.
Proof. We shall prove this for right magnifying element . We consider the following cases.
Case 1. Let be noncyclic.
If is a right magnifying element of , then , where is a proper subhypergroupoid of . Since is noncyclic, it follows that is factorizable because , where .
Case 2. Let be cyclic and .
As is a right magnifying element, , where is a proper subhypergroupoid of . Since is a proper subhypergroupoid of , we get , where . Moreover, for , let . Hence,where , which is a contradiction because is a proper subhypergroupoid of . Now, the proof is completed.
Definition 5. An ordered hypergroupoid (semihypergroup) is said to be generalized factorizable if there exist two different proper subsets and of such that . The pair is called a generalized factorization of , with generalized factors and .
Example 5. (1)Consider the ordered hypergroupoid defined in Example 2. The subsets and then form the generalized factorization of as Thus, can be composed by the parents possessing only and blood groups.(2)Consider the ordered hypergroup defined in Example 4. The subsets and then form the generalized factorization of as
3. 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 -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
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (61772376 and 62072129) and by the National Key R&D Program of China (2019YFA0706402).