y_i(m)\!=(y^0_i(m),y^1_i(m)) \in\Z\times\Z \, \textup{vanishes}\\
 & \textup{except at finitely many} \ (i,m)
\end{aligned}
\right\}.$$ We give the lexicographic order to the set $\Z\times\Z$ of variable exponents. Fix any set of integers $c={(c_{ij})}_{i \neq j \in I}$ such that $$c_{ij}+c_{ji}=1,$$ and set $$A_i(m)={Y_i(m)}^{(0,1)}{Y_i(m+1)}^{(0,1)}
\prod_{j \neq i} {Y_j(m+c_{ji})}^{(0,\langle h_j,\alpha_i\rangle)}.$$

The crystal structure on $\Me$ is defined as follows. For every monomial $M=\prod_{(i,m)\in I\times\Z}{Y_i(m)}^{y_i(m)}$, we set $$\begin{aligned}
\operatorname{\tilde{wt}}(M) &= \sum_i(\sum_m y_i(m))\La_i,\\
\vphit_i(M) &= \textup{max}\{ \sum_{k\le m} y_i(k)
            \,\Big\vert \ m\in\Z \},\\
\vepst_i(M) &= \textup{max}\{ -\sum_{k>m} y_i(k)
            \,\Big\vert \ m\in\Z \}.\end{aligned}$$ Notice that the coefficients of $\operatorname{\tilde{wt}}(M)$ are pairs of integers. In this setting, we have $\vphit_i(M)\geq (0,0)$, $\vepst_i(M)\geq (0,0)$, and $\operatorname{\tilde{wt}}(M)=\sum_i (\vphit_i(M)-\vepst_i(M))\La_i$. Set $$\begin{aligned}
{2}
\operatorname{wt}(M) &= \sum_i(\sum_m y^1_i(m))\La_i,\label{structure1}\\
\vphi_i(M) &= \sum_{k\le m} y^1_i(k) & &\textup{where}\
              \vphit_i(M)= \sum_{k\le m} (y^0_i(k),y^1_i(k)),
              \label{structure2}\\
\veps_i(M) &= -\sum_{k>m} y^1_i(k) & &\textup{where}\
              \vepst_i(M)= -\sum_{k>m} (y^0_i(k),y^1_i(k)).
              \label{structure3}\end{aligned}$$ Then we trivially have $\operatorname{wt}(M)=\sum_i (\vphi_i(M)-\veps_i(M))\La_i$. From the above definition, ${Y_i(m)}^{(0,1)}$ has the weight $\La_i$, and so $A_i(m)$ has the weight $\alpha_i$. We define the action of Kashiwara operators by $$\begin{aligned}
\fit(M) &=
\begin{cases}\label{actions}
0 &\textup{if} \ \vphit_i(M)=(0,0),\\
A_i(m_f)^{-1}M &\textup{if} \ \vphit_i(M)>(0,0),
\end{cases}\\
\eit(M) &=
\begin{cases}\label{actions2}
0 &\textup{if} \ \vepst_i(M)=(0,0),\\
A_i(m_e)M &\textup{if} \ \vepst_i(M)>(0,0).
\end{cases}\end{aligned}$$ Here, $$\begin{aligned}
m_f &=\textup{min}\{m | \vphit_i(M)=
      \sum_{k\le m} 