$\gamma \in \widetilde{B}$.

Suppose that the conditions $(i) - (iii)$ are satisfied for $\widetilde{B}=\{\beta \in B:\ker\beta \in \mathcal{Q}_{m}\}$. We will show that $\mathcal{A} \subseteq \left\langle \O(X,Y),B \right\rangle$. Let $\alpha \in \mathcal{A}\setminus\{\eta\}$. Then there is a partition $P = \{A_{1},A_{2}<\cdots<A_{m}\}\in \mathcal{P}_{m}$ such that $$\alpha = \alpha_{P}=\left(
\begin{array}{cccc}
A_{1} & A_{2} & \cdots  & A_{m} \\
a_{1} & a_{2} & \cdots  & a_{m}\end{array}\right), \mbox{ if } 1 \notin Y \mbox{ or } n \notin Y,$$ or $$\alpha = \alpha_{P}=\left(
\begin{array}{ccccc}
A_{1} & A_{2} & \cdots  & A_{m-1} & A_m\\
a_{2} & a_{3} & \cdots  & a_{m} & a_1
\end{array}
\right), \mbox{ if } 1, n \in Y.$$ Notice that in the latter case $a_1 = 1$ and $a_m = n$.\
Further, from $(i)$ it follows that there is $\beta \in B$ with $\ker\beta =\ker\alpha_P$, i.e. $\beta = \alpha_P$ or $$\beta =\left(
\begin{array}{ccccccc}
A_{1} & A_{2} & \cdots  & A_{m-i+1} & A_{m-i+2} & \cdots  & A_{m} \\
a_{i} & a_{i+1} & \cdots  & a_{m} & a_{1} & \cdots  & a_{i-1}\end{array}\right)$$ for some $i\in \{3,\ldots,m\}$. It is easy to verify that $\alpha_{P}=\beta^{k}\in \left\langle
B\right\rangle$, for a suitable natural number $k$. Hence, $\{\alpha_{P}: P \in \mathcal{P}_{m}\} \subseteq \left\langle\O(X,Y),B \right\rangle$. Further, $\ker \eta \in \mathcal{P}_{m}$, whenever 