$${\tilde{\mathfrak{g}}}= {\tilde{\mathfrak{g}}}_{-1} + {\tilde{\mathfrak{g}}}_0 + {\tilde{\mathfrak{g}}}_1.$$ We define Feigin-Stoyanovsky’s type subspace as $$W(\Lambda) = U({\tilde{\mathfrak{g}}}_1) \cdot v_{\Lambda} \subset L(\Lambda).$$ If we further on set $${\tilde{\mathfrak{g}}}_1^{-}={\tilde{\mathfrak{g}}}_1 \cap {\tilde{\mathfrak{n}}}_{-} = {\mathfrak{g}}_1 \otimes t^{-1} {\mathbb{C}}[t^{-1}]$$ then obviously $$W(\Lambda) = U({\tilde{\mathfrak{g}}}_1^{-}) \cdot v_{\Lambda}.$$ By Poincaré-Birkhoff-Witt theorem, the spanning set of $W(\Lambda)$ consists of monomial vectors $$\begin{aligned}
\label{PBW}
\{ x_{\delta_1}(-n_1) x_{\delta_2}(-n_2) \dots x_{\delta_r}(-n_r)v_{\Lambda} \ \vert \ r \in {\mathbb{Z}}_+, \ n_i \in {\mathbb{N}}, \ \delta \in \Gamma  \ \}.\end{aligned}$$ Elements of this spanning set can be identified with the monomials from $U({\tilde{\mathfrak{g}}}_1) \cong S({\tilde{\mathfrak{g}}}_1)$. Therefore we often refer to the elements of the form $ x_{\delta}(-j)$, $\delta_i \in \Gamma, \ n \in {\mathbb{N}},$ as *the variables, elements or factors of the monomial*.

Monomial ordering {#order}
=================

To reduce the spanning set of $W(\Lambda)$ to a basis, we establish a linear order on the set of monomials. The monomials from $U({\tilde{\mathfrak{g}}}_1)$ can be interpreted as so-called *colored partitions*: $\pi : \{ x_{\delta}(-j) \ \vert \ \delta \in \Gamma, \ j \in {\mathbb{Z}}\} \longrightarrow {\mathbb{Z}}_+$ denotes the monomial $$x(\pi) = x_{\delta_{1}} (-j_1)^{\pi(x_{\delta_{1}}(-j_1))} x_{\delta_{2}}(-j_2)^{\pi(x_{\delta_{2}}(-j_2))} \dots x_{\delta_r}(-j_r)^{\pi(x_{\delta_{r}}(-j_r))}.$$ ${\mathfrak{h}}$-weights $\delta_{i} \in \Gamma$ are referred to as *colors*. For color $\gamma = \epsilon_1 \pm \epsilon_j$ we will define *the oposite color* 