---
tag: K711RM2Z15
kind: definition
name: Monomial order
msc: [13P10, 06A05]
defines: E874NARWPV
uses: [0SDC934R52, 4G25S7PJ22, 0FRXBVVR03, 01CGSJ0GNQ, 0B68KNPGCG]
sources:
  - url: https://en.wikipedia.org/wiki/Monomial_order
---

## Definition

Let $k$ be a field and let $n \ge 1$.
The monomials of the [[polynomial ring|0SDC934R52]] $k[x_1, \ldots, x_n]$ are the products $x^a = x_1^{a_1} \cdots x_n^{a_n}$ with $a \in \N^n$.
A *monomial order* is a relation $>$ on the set of monomials such that:

1. $>$ is a strict [[total order|0FRXBVVR03]].
2. If $x^a > x^b$, then $x^a x^c > x^b x^c$ for every $c \in \N^n$.
3. $>$ is a [[well-order|01CGSJ0GNQ]]: every nonempty set of monomials has a smallest element.

## Remarks

Given clauses 1 and 2, clause 3 is equivalent to $x^a > 1$ for every $a \ne 0$; this uses Dickson's lemma.
Equivalently, a monomial order is a total order on $\N^n$ that is compatible with addition and has $0$ as its least element.
For a nonzero polynomial $f$, the leading monomial $\operatorname{LM}(f)$ is the largest monomial that occurs in $f$, and the leading term $\operatorname{LT}(f)$ is that monomial with its coefficient.

Example: the [[lexicographic order|0B68KNPGCG]]: $x^a > x^b$ if the first nonzero entry of $a - b$ is positive.
It is total, it is compatible since $(a + c) - (b + c) = a - b$, and it is a well-order on $\N^n$.

Non-example: for $n = 1$, order the monomials by $x^a > x^b$ if $a < b$.
This is a total order compatible with multiplication, but $x > x^2 > x^3 > \cdots$ is an infinite descending chain, so clause 3 fails.