Journal of Applied Mathematics

VolumeΒ 2012, Article IDΒ 717698, 8 pages

http://dx.doi.org/10.1155/2012/717698

## The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups

^{1}Department of Applied Mathematics, Hanyang University, Ansan 426-791, Republic of Korea^{2}Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, Republic of Korea^{3}Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487-0350, USA

Received 8 March 2012; Accepted 14 July 2012

Academic Editor: Sazzad H.Β Chowdhury

Copyright Β© 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.

#### 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. IsΓ©ki 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 ββ 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 ββ 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 ββ 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 ββ 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 ββ 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 ββ 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 [7β10] as a general plan for the organization of the subject, with parts to be completed as time and opportunity permits.

#### References

- H. S. Kim and J. Neggers, βThe semigroups of binary systems and some perspectives,β
*Bulletin of the Korean Mathematical Society*, vol. 45, no. 4, pp. 651β661, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - H. F. Fayoumi, βLocally-zero groupoids and the center of $\mathrm{Bin}(X)$,β
*Korean Mathematical Society. Communications*, vol. 26, no. 2, pp. 163β168, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - K. Iséki, βOn BCI-algebras,β
*Mathematics Seminar Notes*, vol. 8, no. 1, pp. 125β130, 1980. View at Google Scholar Β· View at Zentralblatt MATH - K. Iséki and S. Tanaka, βAn introduction to the theory of BCK-algebras,β
*Mathematica Japonica*, vol. 23, no. 1, pp. 1β26, 1978/79. View at Google Scholar - J. Neggers and H. S. Kim, βOn $d$-algebras,β
*Mathematica Slovaca*, vol. 49, no. 1, pp. 19β26, 1999. View at Google Scholar - J. S. Han, H. S. Kim, and J. Neggers, βStrong and ordinary $d$-algebras,β
*Journal of Multiple-Valued Logic and Soft Computing*, vol. 16, no. 3–5, pp. 331β339, 2010. View at Google Scholar - J. Zhan, B. Davvaz, and K. P. Shum, βOn probabilistic $n$-ary hypergroups,β
*Information Sciences*, vol. 180, no. 7, pp. 1159β1166, 2010. View at Publisher Β· View at Google Scholar - J. Zhan and Y. L. Liu, βOn
*f*-derivations of BCI-algebras,β*Mathematica Slovaca*, vol. 49, pp. 19β26, 1999. View at Google Scholar - J. Zhan, S. Sh. Mousavi, and M. Jafarpour, βOn hyperactions of hypergroups,β
*University of Bucharest. Scientific Bulletin A*, vol. 73, no. 1, pp. 117β128, 2011. View at Google Scholar Β· View at Zentralblatt MATH - B. Davvaz and V. Leoreanu,
*Hyperring Theory and Applications*, International Academic Press, Palm Harbor, Fla, USA, 2007.