From N.RUSKUC's paper "On Large Subsemigroups and Finiteness conditions of Semigroups", there is a theorem,
Here large subsemigroup means $S$\ $T$ is finite. In this side "=>" of the proof in the paper, suppose $S$ is finitely generated by the set $A$. Then the set
My question is how can I get that $X$ generates $T$. Of course from the definition of $X$, I know $T$ contains $X$. So the point may be how to get that for any element in T, this element can be presented by X.
Thanks for your assistance.