Research Article | Open Access
Jeong Soon Han, Hee Sik Kim, J. Neggers, "The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups", Journal of Applied Mathematics, vol. 2012, Article ID 717698, 8 pages, 2012. https://doi.org/10.1155/2012/717698
The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups
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 .
The notion of the semigroup was introduced by Kim and Neggers . Fayoumi  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 . The present authors  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.  defined the -fuzzy -ary sub-hypergroups by using a norm and obtained some related properties. Zhan, and Liu  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.  defined the notion of hyperaction of a hypergroup as a generalization of the concept of action of a group. Recently, Davvaz and Leoreanu  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 .
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 ). 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.
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.
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.
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Copyright © 2012 Jeong Soon Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.