Regularity and Green′s Relations on a Semigroup ofTransformations 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‐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.