the form ${\pmb{x}}^{{\pmb{\alpha}}}\triangleq x_1^{\alpha_1} x_2^{\alpha_2}\cdots x_d^{\alpha_d},$ where ${{\pmb{\alpha}}}=(\alpha_1,\alpha_2,\ldots,\alpha_d)\in\mathbb N^d$ is a vector of natural numbers, and $\sum_{i=1}^d \alpha_i$ is called the *degree* of ${\pmb{x}}^{\alpha}$, denoted by $\deg({\pmb{x}}^{{\pmb{\alpha}}})$. A polynomial $f$ of ${{\pmb{x}}}$ is a linear combination of a finite number of monomials over ${\mathbb{K}}$, i.e., $f=\sum_{i=1}^m\lambda_i{\pmb{x}}^{{\pmb{\alpha}}_i},$ where $m$ is the number of distinct monomials of $f$ and $\lambda_i\in{\mathbb{K}}$ is the nonzero coefficient of ${\pmb{x}}^{{\pmb{\alpha}}_i}$, for each $i$. The degree of $f({\pmb{x}})$ is defined as $\deg(f)=\max\{\deg{{\pmb{x}}^{{\pmb{\alpha}}_i}}\mid i=1,2,\cdots,m\}$. Denote by $\mathcal{M}[{\pmb{x}}]$ the set of monomials of ${\pmb{x}}$ and ${\mathbb{K}}[{\pmb{x}}]$ the polynomial ring of ${\pmb{x}}$ over ${\mathbb{K}}$. The degree of a finite set $X\subseteq{\mathbb{K}}[{\pmb{x}}]$ is defined as $\deg(X)=\max\{\deg(f)\mid f\in X\}$.

we introduce the lexicographic order for monomials: ${\pmb{x}}^{{\pmb{\alpha}}}\prec{\pmb{x}}^{\pmb{\beta}}$ if there exists $1\leq i\leq d$ such that $\alpha_i<\beta_i$ and $\alpha_j=\beta_j$ for all $1\leq j < i$. For every polynomial $f\in{\mathbb{K}}[{\pmb{x}}]$ we write its leading monomial (i.e., the greatest monomial under $\prec $) as $\operatorname{lm}(f)$. For any $n\in{\mathbb{N}}$, a set of monomials $M$ is called *$n$-compressed*, if for any $m\in M$, $\deg(m)=n$, then $\{m'\mid\deg(m')=n,m'\prec m\}\subseteq M.$

1.  A nonempty subset $I\subseteq{\mathbb{K}}[{\pmb{x}}]$ is called an ideal if $f,\ g\in I\Rightarrow f-g\in I$, and $f\in I,\ h\in{\mathbb{K}}[{\pmb{x}}]\Rightarrow fh\in I$.

2.  Let $P$ be a nonempty subset of 