$\T(X,Y)$ was studied in [@FHQS1] by Fernandes, Honyam, Quinteiro, and Singha. The authors found that $\rank\O(X,Y)=\left(
\begin{array}{c}
n-1 \\
m-1\end{array}\right) +\left\vert Y^{\#}\right\vert,$ where $Y^{\#}$ denotes the set of all $y\in Y$ with one of the following properties: (i) $y$ has no successor in $X$; (ii) $y$ is no successor of any element in $X$; (iii) both the successor of $Y$ and the element whose successor is $y$ belong to $Y$. Moreover, the regularity and the rank of the semigroup $\OP(X,Y)$ were studied by the same authors in [@FHQS2]. They showed that $\rank\OP(X,Y)=\left(
\begin{array}{c}
n \\
m
\end{array}\right)$. In [@TK], Tinpun and Koppitz studied the relative rank of $\T(X,Y)$ modulo $\O(X,Y)$ and proved that $\rank(\T(X,Y):\O(X,Y))=
S(n,m)-
\left(
\begin{array}{c}
n-1 \\
m-1
\end{array}
\right) + a$, where $a \in \{0,1\}$ depending on the set $Y$. In this paper, we determine the relative rank of $\OP(X,Y)$ modulo $\O(X,Y)$ as well as the relative rank of $\T(X,Y)$ modulo $\OP(X,Y)$.

Let $\alpha \in \T(X,Y)$. The kernel of $\alpha$ is the equivalence relation $\ker\alpha $ with $(x,y)\in \ker\alpha $ if $x\alpha
=y\alpha$. It corresponds uniquely to a partition on $X$. This justifies to regard $\ker \alpha $ as a partition on $X$. We will call a 