Regularity and Green′s Relations on a Semigroup ofTransformations with Restricted Range
Jintana Sanwong, Worachead Sommanee · International Journal of Mathematics and Mathematical Sciences · 2008
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.