$B=\overline{B}\cup \widehat{B}$, where $\{\ker
\beta :\beta \in B\setminus \widehat{B}\}\supseteq\mathcal{P}_{m}$, $|B\setminus\widehat{B}| = |\overline{B}| = |\mathcal{P}_{m}|$ and $\eta |_{Y}\in
\left\langle \beta |_{Y}:\beta \in B\right\rangle$ but $\eta |_{Y}\notin
\left\langle \beta |_{Y}:\beta \in B\setminus \{\gamma \}\right\rangle $ for any $\gamma \in \widehat{B}$.

  \
In particular, for the relative generating sets of minimal size we have

$B\subseteq \OP(X,Y)$ is a relative generating set of $\OP(X,Y)$ modulo $\O(X,Y)$ of minimal size if and only if $|\widetilde{B}| = 1$ if $1,n \in Y$ and $\widetilde{B} = \emptyset$, otherwise.

The relative rank of $\T(X,Y)$ modulo $\OP(X,Y)$
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In this section, we determine the relative rank of $\T(X,Y)$ modulo $\OP(X,Y)$ and characterize all minimal relative generating sets of $\T(X,Y)$ modulo $\OP(X,Y)$. Since $\O(X,Y)\leq \OP(X,Y)$, we see immediately that $\rank(\T(X,Y):\OP(X,Y))\leq S(n,m)-\left(
\begin{array}{c}
n-1 \\
m-1\end{array}\right) +1$. First, we state a sufficient condition for a set $B\subseteq
\T(X,Y)$ to be a relative generating set of $\T(X,Y)$ modulo $\OP(X,Y)$.

\[le7\] Let $B\subseteq \T(X,Y)$. If $\mathcal{R}_{m}\subseteq \{\ker \beta :\beta\in B\}$ and $\S(Y)\subseteq \left\langle \{\beta|_{Y}:\beta \in B\},\eta|_{Y}\right\rangle $ then $\left\langle \OP(X,Y),B\right\rangle =\T(X,Y)$.

Let $\gamma \in \T(X,Y)$ with $\rank \gamma =k \leq m$. We will consider two cases.

*Case 1.* Suppose that $\ker \gamma \in \mathcal{R}_{k}$. Then $\ker \gamma$ contains a non-convex set which cannot 