: A)$, is the minimal cardinality of a set $B\subseteq S$ such that $A \cup B$ generates $S$. It follows immediately from the definition that $\rank (S : \emptyset) = \rank S$, $\rank ( S : S) = 0$, $\rank (S : A ) = \rank (S : \langle A \rangle )$ and $\rank (S : A) = 0$ if and only if $A$ is a generating set for $S$. The relative rank of a semigroup modulo a suitable set was first introduced by Ruškuc in [@Ruskuc] in order to describe the generating sets of semigroups with infinite rank. In [@HRH], Howie, Ruškuc, and Higgins considered the relative ranks of the monoid $\T(X)$ of all full transformations on $X$, where $X$ is an infinite set, modulo some distinguished subsets of $\T(X)$. They showed that $\rank (\T(X) : \S(X)) = 2$, $\rank (\T(X) : \E(X)) = 2$ and $\rank (\T(X) : J) = 0$, where $\S(X)$ is the symmetric group on $X$, $\E(X)$ is the set of all idempotent transformations on $X$ and $J$ is the top $\mathcal{J}$-class of $\T(X)$, i.e. $J=\{\alpha\in\T(X) : |X\alpha|=|X|\}$. But also if the rank is finite, the relative rank gives information about the generating sets. In 