International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Article ID 12757, 10 pages
On transformation semigroups which are -semigroups
Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
Received 23 December 2005; Revised 6 June 2006; Accepted 22 June 2006
Copyright © 2006 S. Nenthein and Y. Kemprasit. 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.
A semigroup whose bi-ideals and quasi-ideals coincide is called a
-semigroup. The full transformation semigroup on a set
and the semigroup of all linear transformations of a vector
space over a field into itself are denoted, respectively, by and . It is known that every regular semigroup is a
-semigroup. Then both and are -semigroups.
In 1966, Magill introduced and studied the subsemigroup of , where and . If is a subspace of , the subsemigroup of will be defined analogously. In this paper, it is shown that is a -semigroup if and only if , , or , and is a -semigroup if and only if (i) , (ii) , or (iii) , , and .