International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 794013, 11 pages
doi:10.1155/2008/794013

Regularity and Green's relations on a semigroup of transformations with restricted range.

Jintana Sanwong1 and Worachead Sommanee2

1Department of Mathematics, Chiang Mai University, Chiangmai 50200, Thailand
 2Department of Mathematics, Chiangmai Rajabhat University, Chiangmai 50300, Thailand

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.