Regularity and abundance on semigroups of transformations preserving an equivalence relation on an invariant set

Abstract

<abstract><p>Let $ T(X) $ be the full transformation semigroup on a nonempty set $ X $. For an equivalence relation $ E $ on $ X $ and a nonempty subset $ Y $ of $ X $, let</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \overline{S}_E(X, Y) = \{ \alpha \in T(X) : \forall x, y \in Y, (x, y) \in E \Rightarrow (x \alpha, y \alpha) \in E, x \alpha, y \alpha \in Y \}. $\end{document} </tex-math></disp-formula></p> <p>Then $ \overline{S}_E(X, Y) $ is a subsemigroup of $ T(X) $ consisting of all full transformations that leave $ Y $ and the equivalence relation $ E $ on $ Y $ invariant. In this paper, we show that $ \overline{S}_E(X, Y) $ is not regular in general and determine all its regular elements. Then we characterize relations $ \mathcal{L} $, $ \mathcal{L}^* $, $ \mathcal{R} $ and $ \mathcal{R}^* $ on $ \overline{S}_E(X, Y) $ and apply these characterizations to obtain the abundance on such semigroup.</p></abstract>