Journal of Applied Mathematics
Volume 2012 (2012), Article ID 717698, 8 pages
http://dx.doi.org/10.1155/2012/717698
Research Article

The Hypergroupoid Semigroups as Generalizations of the Groupoid Semigroups

Jeong Soon Han,1 Hee Sik Kim,2 and J. Neggers3

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

Received 8 March 2012; Accepted 14 July 2012

Academic Editor: Sazzad H. Chowdhury

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.

Abstract

We introduce the notion of hypergroupoids ( 𝐻 B i n ( 𝑋 ) , □ ) , and show that ( 𝐻 B i n ( 𝑋 ) , □ ) is a super-semigroup of the semigroup ( B i n ( 𝑋 ) , □ ) via the identification 𝑥 ↔ { 𝑥 } . We prove that ( 𝐻 B i n ∗ ( 𝑋 ) , ⊖ , [ ∅ ] ) is a 𝐵 𝐶 𝐾 -algebra, and obtain several properties of ( 𝐻 B i n ∗ ( 𝑋 ) , □ ) .