of $M$ show the property $\ge (0,1)$. Thus $\fit M$ given in  and also are in $\M(\infty)$. So the set $\M(\infty)$ is closed under the above operator $\fit$.

As we can see in equations  to , for each $M\in\M(\infty)$, $\fit M$ can also be expressed in form $MA_i(m)^{-1}$. To show that this operation is just another interpretation of the Kashiwara operator $\fit$ given on $\Me$, restricted to $\M(\infty)$, it is enough to show that $m_f$ defined in  for each $M$ is equal to $m$ of $MA_i(m)^{-1}$ given in equations  to .

Given a monomial $M\in\M(\infty)$, we can express it in the following two forms. $$\begin{aligned}
      M=& X_1(-1)^{(2,-b^{-1}_2-b^{-1}_3-b^{-1}_0-
                      b^{-1}_{\bar3}-b^{-1}_{\bar2}-b^{-1}_{\bar1})}
          X_2(-1)^{(0,b_2^{-1})}
          X_3(-1)^{(0,b_3^{-1})}\label{eq22}\\
        & \cdot
          X_0(-1)^{(0,b_0^{-1})}
          X_{\bar3}(-1)^{(0,b_{\bar3}^{-1})}
          X_{\bar2}(-1)^{(0,b_{\bar2}^{-1})}
          X_{\bar1}(-1)^{(0,b_{\bar1}^{-1})}\notag\\
        & \cdot
          X_2(-2)^{(1,-b_3^{-2})}
          X_3(-2)^{(0,b_3^{-2})}\notag\\
       =&{Y_1(-1)}^{(1,-b^{-1}_2-b^{-1}_3-b^{-1}_0-b^{-1}_{\bar3}
                 -b^{-1}_{\bar2}-b^{-1}_{\bar1}+3b^{-2}_3)}\label{eq23}\\
  &\cdot {Y_1(0)}^{(0,b^{-1}_0-b^{-1}_2+2b^{-1}_3)}
   {Y_1(1)}^{(0,-b^{-1}_0-2b^{-1}_{\bar 3}+b^{-1}_{\bar2})}
   {Y_1(2)}^{(0,-b^{-1}_{\bar1})}\notag\\
  &\cdot {Y_2(-2)}^{(1,-b^{-2}_3)}
         {Y_2(-1)}^{(0,b^{-1}_2-b^{-2}_3)}
         {Y_2(0)}^{(0,b^{-1}_{\bar3}-b^{-1}_3)}
         {Y_2(1)}^{(0,-b^{-1}_{\bar2})}\notag.\end{aligned}$$ If the $1$-signature of $M$ contains at least one $0$ and $X$ is the component corresponding to the left-most $0$ in the $1$-signature of $M$, then we can obtain $$m_f=\textup{min}\{j\in\Z\ \vert\
    \textup{max}\{ \sum_{k\le j} y_1(k)\}\}=
\begin{cases}
1 &\text{if $X={X_{\bar 2}(-1)}^{(0,b_{\bar 2}^{-1})}$},\\
0 &\text{if $X={X_0(-1)}^{(0,b_0^{-1})}$},\\
0 &\text{if $X={X_3(-1)}^{(0,b_3^{-1})}$},
\end{cases}$$ where $y_1(k)$ is the exponent of $Y_1(k)$ appearing in $M$ given by expression . If the $2$-signature of $M$ contains 