Regularity and Finiteness Conditions on Transformation Semigroups with Invariant Sets

Abstract

Let $ X $ be a nonempty set and $ T(X) $ denote the semigroup of transformations from X to itself under the composition of functions. For a fixed nonempty subset $ Y $ of $ X $, let $$ S(X,Y) = \{\alpha \in T(X) : Y\alpha \subseteq Y\}. $$ Then $ S(X,Y) $ is a semigroup of total transformations of $ X $ which leave a subset $ Y $ of $ X $ invariant. In this paper, we characterize coregular elements of $ S(X,Y) $ and give necessary and sufficient conditions for $ S(X,Y) $ to be coregular. Moreover, we study some properties of regularity on $ S(X,Y) $ and give necessary and sufficient conditions for $ {S(X,Y)} $ to be left regular, right regular, completely regular, intra-regular and directly finite.