y_i(k) \}
    =\textup{min}\{m | \vepst_i(M)=
      -\sum_{k>m} y_i(k) \},\label{emf}\\
m_e &=\textup{max}\{m | \vphit_i(M)=
      \sum_{k\le m} y_i(k) \}
    =\textup{max}\{m | \vepst_i(M)=
      -\sum_{k>m} y_i(k) \}.\label{eme}\end{aligned}$$ Note that $y_i(m_f)>(0,0)$, $y_i(m_f+1)\le (0,0)$, $y_i(m_e+1)<(0,0)$, and $y_i(m_e)\ge (0,0)$.

The Kashiwara operators, together with the maps $\vphi_i$, $\veps_i$ $(i\in I)$, $\operatorname{wt}$, define a crystal structure on the set $\Me$ [@lee].

The set of monomials $\prod_{(i,m)\in
I\times\Z}{Y_i(m)}^{(y^0_i(m),y^1_i(m))}$ of $\Me$ with $y^0_i(m)=0$ is exactly the Nakajima monomial set $\M$ if we identify ${Y_i(m)}^{(0,y^1_i(m))}\in\Me$ with ${Y_i(m)}^{y^1_i(m)}\in\M$. The crystal structure on $\M$, introduced in [@MR1988989], is compatible with that on $\Me$ under this identification. The crystal $\M$ is a subcrystal of $\Me$.

Restriction of the following theorem to just the monomials of $\M$ appears in [@MR1988989].

[@lee]\[mla\] If a monomial $M\in\Me$ of $\operatorname{\tilde{wt}}(M)=\sum_i(0,p_i)\La_i$, where each $p_i$ is a nonnegative integer, satisfies $\eit(M)=0$ for all $i\in I$, then the connected component of $\Me$ containing $M$ is isomorphic to $\B(\sum_i p_i\La_i)$ as a $\uq(\g)$-crystal. Conversely, given any subset of $\Me$ isomorphic to $\B(\sum_i p_i\La_i)$, there exists an element $M$ in the subset such that $\operatorname{\tilde{wt}}(M)=\sum_i(0,p_i)\La_i$ and $\eit(M)=0$ for all $i\in I$.

The following is a conjecture on the crystal $\B(\infty)$ introduced in [@lee] and stated for all symmetrizable Kac-Moody algebras. Its converse is known 