3 Monomial Ideal
3.1 Orderings on the Monomials in \(k[x_1,\ldots ,x_n]\)
Let \(\iota \) be an index set and \(s \subset \iota \) a finite subset. For each \(i \in s\), let \(h_i \in k[x_1,\dots ,x_n]\). Then the following inequality holds:
where the \(\max \) is taken with respect to the monomial order.
Let \(M = \max _{i \in s} \{ \operatorname {multideg}(h_i) \} \). Any monomial \(x^b\) appearing in the sum \(\sum _{i \in s} h_i\) must be a monomial in at least one of the summands, say \(h_{i_0}\) for some \(i_0 \in s\). By definition, the multidegree of any such term is bounded by the multidegree of the polynomial it belongs to, so \(b \le \operatorname {multideg}(h_{i_0})\). Also by definition, \(\operatorname {multideg}(h_{i_0}) \le M\). Therefore, \(b \le M\) for any monomial \(x^b\) in the sum. This implies that the multidegree of the sum itself cannot exceed \(M\).
3.2 Monomial Ideals and Dickson’s Lemma
An ideal \(I \subseteq k[x_1,\dots ,x_n]\) is called a monomial ideal if there is a subset \(A \subseteq \mathbb {Z}_{\ge 0}^n\) (possibly infinite) such that \(I\) consists of all polynomials which can be written as finite sums of the form
In this case we write
Let \(I=\langle x^\alpha \mid \alpha \in A\rangle \) be a monomial ideal in \(k[x_1,\dots ,x_n]\). For a polynomial \(f=\sum _{\gamma } c_\gamma x^\gamma \), write \(\operatorname {supp}(f)=\{ \gamma \mid c_\gamma \neq 0\} \). Then
equivalently, every monomial term \(x^\gamma \) appearing in \(f\) is divisible by some generator \(x^\alpha \) with \(\alpha \in A\).
By definition, \(I=\langle x^\alpha \mid \alpha \in A\rangle \) is the ideal generated by the set of monomials \(\{ x^\alpha :\alpha \in A\} \).
(\(\Rightarrow \)) Assume \(f\in I\). Then \(f\) is a finite \(k[x_1,\dots ,x_n]\)-linear combination of the generators:
Expanding each \(h_i\) into monomials, \(h_i=\sum _j c_{i,j}x^{\beta (i,j)}\), we get
Hence every exponent \(\gamma \) in \(\operatorname {supp}(f)\) must be of the form \(\gamma =\beta (i,j)+\alpha (i)\) for some \((i,j)\) with \(c_{i,j}\neq 0\), so \(\alpha (i)\le \gamma \). This shows: for each \(\gamma \in \operatorname {supp}(f)\) there exists \(\alpha \in A\) with \(\alpha \le \gamma \).
(\(\Leftarrow \)) Conversely, assume that for every \(\gamma \in \operatorname {supp}(f)\) there exists \(\alpha (\gamma )\in A\) such that \(\alpha (\gamma )\le \gamma \). Then for each such \(\gamma \) we can write \(\gamma =\alpha (\gamma )+\delta (\gamma )\) with \(\delta (\gamma )\in \mathbb Z_{\ge 0}^n\), and
since \(x^{\alpha (\gamma )}\) is a generator and \(I\) is an ideal. Summing over all \(\gamma \in \operatorname {supp}(f)\) gives \(f\in I\).
Let \((\mathbb {N}^n, \le ')\) be the direct product of \(n\) copies of the natural numbers \((\mathbb {N}, \le )\) with their natural ordering. Then \((\mathbb {N}^n, \le ')\) is a Dickson partially ordered set. More explicitly, every subset \(S \subseteq \mathbb {N}^n\) has a finite subset \(B\) such that for every \((m_1, \dots , m_n) \in S\), there exists \((k_1, \dots , k_n) \in B\) with
By Proposition 4.42, a quasi-ordered set has the Dickson property if and only if it is well-quasi-ordered. Thus, it suffices to show that for any sequence in \(\mathbb {N}^n\) there exist indices \(i{\lt}j\) such that \(\mathbf{x}_i \le ' \mathbf{x}_j\).
Let \(\{ \mathbf{x}_k\} _{k\in \mathbb {N}}\) be an arbitrary sequence in \(\mathbb {N}^n\). By a standard result (\((\mathbb {N}^n,\le ')\) is partially well-ordered), there exists a strictly increasing map \(g:\mathbb {N}\to \mathbb {N}\) such that
Set \(i \coloneqq g(0)\) and \(j \coloneqq g(1)\). Then \(i{\lt}j\) and, by monotonicity, \(\mathbf{x}_i \le ' \mathbf{x}_j\). Hence \((\mathbb {N}^n,\le ')\) is well-quasi-ordered, and therefore has the Dickson property by Proposition 4.42.
Let \(I = \langle x^{\alpha } | \alpha \in A \rangle \subseteq k[x_1, \ldots , x_n]\) be a monomial ideal. Then \(I\) can be written in the form \(I = \langle x^{\alpha (1)}, \ldots , {\alpha (s)} \rangle \), where \(\alpha (1), \ldots , \alpha (s) \in A\). In particular, \(I\) has a finite basis.