Abstract
We introduce the notion of hypergroupoids , and show that is a super-semigroup of the semigroup via the identification . We prove that is a -algebra, and obtain several properties of .
1. Introduction
The notion of the semigroup was introduced by Kim and Neggers [1]. Fayoumi [2] introduced the notion of the center in the semigroup of all binary systems on a set , and showed that if , then implies . Moreover, she showed that a groupoid if and only if it is a locally zero groupoid. Y. Imai and K. Iséki introduced two classes of abstract algebras: -algebras and -algebras [3, 4]. Neggers and Kim introduced the notion of -algebras which is another useful generalization of -algebras, and then investigated several relations between -algebras and -algebras as well as several other relations between -algebras and oriented digraphs [5]. The present authors [6] defined several special varieties of -algebras, such as strong -algebras, (weakly) selective -algebras, and pre--algebras, discussed the associative groupoid product , where . They showed that the squared algebra of a pre--algebra is a strong -algebra if and only if is strong.
Zhan et al. [7] defined the -fuzzy -ary sub-hypergroups by using a norm and obtained some related properties. Zhan, and Liu [8] introduced the notion of -derivation of a -algebras. They gave some characterizations of a -semisimple -algebras by using the idea of a regular -derivation. Zhan et al. [9] defined the notion of hyperaction of a hypergroup as a generalization of the concept of action of a group. Recently, Davvaz and Leoreanu [10] published a beautiful book, Hyperring Theory and Applications, and provided useful information on the theory of the hypertheory.
In this paper we introduce the notion of hypergroupoids , and show that is a super-semigroup of the semigroup via the identification . We prove that is a -algebra, and obtain several properties of .
2. Preliminaries
Given a nonempty set , we let the collection of all groupoids , where is a map and where is written in the usual product form. Given elements and of , define a product “” on these groupoids as follows: where for any . Using the notion, H. S. Kim and J. Neggers showed the following theorem.
Theorem 2.1 (see [1]). is a semigroup, that is, the operation “” as defined in general is associative. Furthermore, the left zero semigroup is an identity for this operation.
3. Hypergroupoid Semigroups
Instead of a groupoid on , we may also consider a hypergroupoid on , where is a hyperproduct with , the set of all non-empty subsets of . We denote the set of all hypergroupoids on by , that is, The product “” discussed in can be generalized in as follows: given , for any , If we identify with , then we have an inclusion: and thus for , we have and hence also via this identification.
If , then for the groupoid , we have hence in a natural way. Similarly, given a hypergroupoid , is defined by .
Given hypergroupoids , we let . Then, for any , we have
Suppose that and are groupoids and that we determine the following: via the identification . Hence is the same as a product of groupoids or as a product of hypergroupoids.
It can be shown that is an injection (an into homomorphism) via the identification and the associated identification .
Example 3.1. Let and for any , let denote the undirected line segment connecting with . Then and . Let . Then for any . Since , for any . Since , . We claim that . If , then for some and . Since , for some , which shows that . This proves that , that is, is an idempotent hypergroupoid in .
Theorem 3.2. is a supersemigroup of the semigroup via the identification .
Proof. Suppose that and are hypergroupoids and let and . Then for any , we have and . Let . Then and hence we obtain the following If we let , then and hence for any . Let . Then This proves that , that is, is a semigroup.
Proposition 3.3. The left-zero-semigroup , that is, for any , is an identity of the semigroup .
Proof. Let be a left-zero-semigroup. Then . By the identification , we have . Given , let . Then for any , we have
that is, . This proves that is a left identity on .
Similarly, if we let , then for any ,
that is, . This proves that is a right identity on .
Given an element , , that is, . We extend to as by , where . In particular, This produces a mapping . Let . Then for any . Since , we have Since via the identification , we obtain where in . We claim that is a homomorphism. In fact, .
Given , we may order it according to the rule We define a mapping by for all . If we let , then is the minimal element of .
Proposition 3.4. Let and . If , then .
Proof. If , then for any . It follows that , proving that .
Proposition 3.5. Let . If , then , that is, is an antichain in .
Proof. If , then for any . It follows that for any , proving that .
4. -Algebras on
In this section we discuss -algebras on by introducing a binary operation as follows: given hypergroupoids , we define a binary operation “” by where for any .
Theorem 4.1. is a -algebra.
Proof. For any , since for any , we have .
Given , since for any , we have .
Assume that . Then for any , which shows that for any , that is, .
Given , since for any , we obtain .
Given , since for any , we obtain . This proves the theorem.
5. Several Properties on
In this section, we discuss some properties on .
Proposition 5.1. The product “” is order-preserving, that is, if , then .
Proof. Let in . If we let and , then for any , proving that .
We define a mapping by for all . Then is the maximal element of . Given , if we let , then for any .
Proposition 5.2. If , then .
Proof. Let . Then, for any , we have proving that .
Given , we define a hypergroupoid by , for any . We call it the complementary hypergroupoid of .
For example, if is a group, then , where . It follows that and for any .
A hypergroupoid is said to be a complementary -algebra if there exists such that (i) ; (ii) ; (iii) implies , for any .
The following proposition can be easily seen.
Proposition 5.3. Given , is a -algebra if and only if is a complementary -algebra.
Example 5.4. Let be the set of all real numbers and be a mapping. Define a map by . Then be a hypergroupoid for which has a midpoint where .
In particular, let for any and let . Then = = = , an interval of length , where implies , corresponding to 0 in the identification.
A hypergroupoid is said to be left inclusive if for any .
Note that the only left inclusive hypergroupoid which is a groupoid is the left-zero-semigroup. In fact, let be a left inclusive hypergroupoid which is a groupoid. Then for any . It follows that for any , that is, is a left-zero-semigroup.
Proposition 5.5. The left inclusive hypergroupoids on relative to the product “” on form a subsemigroup of .
Proof. Let be left inclusive hypergroupoids and let . Then for any . Since is left inclusive, , and hence for any . Moreover, is left inclusive implies that , which proves that .
Proposition 5.6. Let in . If is left inclusive, then is also left inclusive.
Proof. Let . Then for any . Since is left inclusive, we have , proving the proposition.
Proposition 5.6 means that the collection of all left inclusive hypergroupoids is a filter in the poset .
A hypergroupoid is said to be left-self-avoiding if for any .
Proposition 5.7. The complementary hypergroupoid of a left inclusive hypergroupoid is left-self-avoiding.
Proof. Let be the complementary hypergroupoid of a left inclusive hypergroupoid . Then for any . Since is left inclusive, for any , and hence , proving the proposition.
Proposition 5.8. The complementary hypergroupoid of a left-self-avoiding hypergroupoid is left inclusive.
Proof. Straightforward.
Proposition 5.9. Let where is left inclusive and is left-self-avoiding. Then is left-self-avoiding.
Proof. Let be a left-self-avoiding hypergroupoid. Then is left inclusive by Proposition 5.8. It follows that . This means that for any and where . Since is left inclusive, . Hence , proving that is left-self-avoiding.
6. Conclusion
In this paper we have introduced the notion of hypergroupoids as a generalization of groupoids in a manner analogous to the introduction of the notion of hypergroups as a generalization of the notion of groups. Since the semigroup can still benefit from more detailed investigation it follows that the same is even more true for . In the latter case one must rely on proper adaptations obtained from and certainly on results obtained from studies on hypergroupoids available in the literature [7–10] as a general plan for the organization of the subject, with parts to be completed as time and opportunity permits.