For simplicity, from now on, we take the set $C=(c_{ij})_{i\neq j \in I}$ to be $c_{12}=1$ and $c_{21}=0$. Then for $m\in\Z$, we have $$\begin{cases}
A_1(m)={Y_1(m)}^{(0,1)}{Y_1(m+1)}^{(0,1)}{Y_2(m)}^{(0,-1)},\\
A_2(m)={Y_2(m)}^{(0,1)}{Y_2(m+1)}^{(0,1)}{Y_1(m+1)}^{(0,-3)}.
\end{cases}$$

The set we define below was originally obtained by applying Kashiwara actions $\fit$ continuously on the single element ${Y_1(-1)}^{(1,0)}{Y_2(-2)}^{(1,0)}\in\Me$. This choice of starting monomial will allow us to relate monomials of the set defined below to tableaux in $\T(\infty)$ naturally.

Consider elements of $\Me$ having the form $$\label{extend}
\begin{aligned}
      M=&{Y_1(-1)}^{(1,a_1^{-1})}{Y_1(0)}^{(0,a_1^0)}
        {Y_1(1)}^{(0,a_1^1)}{Y_1(2)}^{(0,a_1^2)}\\
        &\cdot {Y_2(-2)}^{(1,a_2^{-2})}{Y_2(-1)}^{(0,a_2^{-1})}
        {Y_2(0)}^{(0,a_2^0)}{Y_2(1)}^{(0,a_2^1)}
\end{aligned}$$ with conditions

1.  $(a_2^{-2}-a_2^{-1})$, $a_2^1$, $a_1^2$, $a_2^{-2}\le 0$,

2.  $(a_1^{-1}-a_1^1-a_1^2)+(2a_2^{-2}+a_2^{-1}-a_2^0-2a_2^1)=0$ and\
    $(a_1^{-1}+a_1^0-a_1^2)+(a_2^{-2}+2a_2^{-1}+a_2^0-a_2^1)=0$,

3.  $(a_1^0+a_2^{-1}-a_2^{-2}), (-a_1^1-a_2^1)\in 2\Z_{\ge 0}$ or\
    $(a_1^0+a_2^{-1}-a_2^{-2}), (-a_1^1-a_2^1)\in \Z_{\ge 0}$ and odd.

Specifically, in case of $a_i^j=0$ for all $i,j$, we have $$\label{hi9}
      M={Y_1(-1)}^{(1,0)}{Y_2(-2)}^{(1,0)}.$$ We denote by $\M(\infty)$ the set of all monomials of these form and by $M_{\infty}$ the monomial of .

Actually, as we will become apparent later, this set $\M(\infty)$ is closed and connected under Kashiwara operators. Figure \[fig:1\] is the top part of monomial set $\M(\infty)$.

in l:0.065 w:0.03 t:F h:C v:C em 1.9 (-1 -1.4)(-1 -1)(1 -1.4)(1 -1)(-3.2 -4.5)(1 -1)(-4.9 -4.5)(-1 -1)(3.2 -4.5)(-1 -1)(5.2 -4.5)(1 -1)(-8 -7.5)(-1 -1)(-6.5 -7.5)(1 -2.7)(-2 -7.5)(-1 -1)(-0.3 -7.5)(1 -2.7)(4 -7.5)(-1 -1)(5 -7.5)(0.5 -1)(6 -10.1)(0.25 -0.5)(8 -7.5)(-1 