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

Abstract

Let T(X) be the full transformation semigroup on the set X and let T (X), Y) = {α ∈ T (X) : X α ⊆ 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-semigroupis 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. Copyright © 2008 J. Sanwong and W. Sommanee.