Name the digraph of these transformation subsemigroups

Question

I am trying to track down the name of this structure and some references.

You take all members of the transformation semigroup on $n$ elements, $T_n$.

For two members $x, y\in T_n$: if $x$ is in the subsemigroup generated by $y$ then you put an arrow from $y$ to $x$. You would read this as $x$ is in the subsemigroup generated by $y$.

What is the name of this digraph? References would be great.

Here is the graph for $T_{3}$.

Answer 1

This is the partial order under containment on the monogenic (generated by 1 element) subsemigroups of the full transformation semigroup $T_n$. A digraph is just a binary relation and a partial order is a special type of binary relation.

This is not a property which is specific to transformation semigroups (your definition didn't involve transformation semigroups), but to semigroups in general.

You could calculate this partial order using GAP for some small values of $n$.