A note on the rank of semigroups. (Q711607)

Let \(\mathcal A\) be a semigroup of transformations of a finite set \(X\), \(Y\subseteq X\) and \(\mathcal B\) a transitive semigroup of transformations of the set \(Y\). If \(f(X)\subseteq Y\) for some \(f\in\mathcal A\) and each element of \(\mathcal B\) is the restriction to \(Y\) of an element of \(\mathcal A\), then the rank \(\min_{f\in\mathcal A}|f(X)|\) of the semigroup \(\mathcal A\) divides the rank of the semigroup \(\mathcal B\).