International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 794013, 11 pages
doi:10.1155/2008/794013
Research Article
Regularity and Green's Relations on a Semigroup of
Transformations with Restricted Range
1Department of Mathematics, Chiang Mai University, Chiangmai 50200, Thailand
2Department of Mathematics, Chiangmai Rajabhat University, Chiangmai 50300, Thailand
Received 14 May 2008; Accepted 18 September 2008
Academic Editor: Robert Redfield
Copyright © 2008 Jintana Sanwong and Worachead Sommanee. 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
Let T(X) be the full transformation semigroup on the set X and let T(X,Y)={α∈T(X):Xα⊆Y}.
Then T(X,Y) is a sub-semigroup of T(X) determined by a nonempty subset Y of X. In this paper, we give a necessary and sufficient condition for T(X,Y) to be regular. In the case that T(X,Y) is not regular, the largest regular sub-semigroup is obtained and this sub-semigroup
is shown to determine the Green's relations on T(X,Y). Also, a class
of maximal inverse sub-semigroups of T(X,Y) is obtained.