Let 𝒯_X be the full transformation semigroup on the nonempty set X. We fix a nonempty subset Y of X and consider the semigroup of transformations that leave Y invariant, and endow it with the so-called natural partial order. Under this partial order, we determine when two elements of S(X,Y ) are related, find the elements which are compatible and describe the maximal elements, the minimal elements and the greatest lower bound of two elements. Also, we show that the semigroup S(X,Y ) is abundant.

In this short note, we describe all the elements in the semigroup $$\begin{eqnarray}S(X,Y)=\{f\in {\mathcal{T}}_{X}:f(Y)\subseteq Y\}\end{eqnarray}$$ which are left compatible with respect to the so-called natural partial order. This result corrects an error in a paper by Sun and Wang [‘Natural partial order in semigroups of transformations with invariant set’, Bull. Aust. Math. Soc.87 (2013), 94–107].