modulo $\OP(X,Y)$.

Conversely, let $B$ be a minimal relative generating set of $\T(X,Y)$ modulo $\OP(X,Y)$. We have $\mathcal{R}_{m}\subseteq
\{\ker \beta :\beta \in B\}$ and $\S(Y)\subseteq \left\langle \{\beta
|_{Y}:\beta \in B\},\eta |_{Y}\right\rangle $ by Lemma \[le9\] and Lemma \[le8\], respectively. Then there exists a set $\widetilde{B}\subseteq B$ with $|B\setminus \widetilde{B}| = |\mathcal{R}_{m}|$ and $\mathcal{R}_{m}\subseteq \{\ker \beta :\beta \in (B\setminus \widetilde{B})\}$. For the set $\widetilde{B}$, the conditions $(i)$ and $(ii)$ are satisfied. Assume now that there is $\gamma \in B$ with $\ker \gamma \in \{\ker \beta :\beta
\in \widetilde{B}\}$ such that $\S(Y)\subseteq \left\langle \{\beta
|_{Y}:\beta \in B\setminus \{\gamma\}\},\eta |_{Y}\right\rangle$. Then because of $\mathcal{R}_{m}\subseteq \{\ker \beta :\beta \in (B\setminus \{\gamma\})\}$, the set $B\setminus \{\gamma)$ is a relative generating set of $\T(X,Y)$ modulo $\OP(X,Y)$ by Lemma \[le7\]. This contradicts the minimality of $B$. Consequently, $(iii)$ is satisfied.

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

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

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Catarino P. M., Higgins P. M., The monoid of orientation-preserving mappings on a chain, Semigroup Forum, 58 (1999), 190–206.

Dimitrova I., Fernandes V. H., Koppitz J., A note on generators of the endomorphism semigroup 