Monomial order

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: $x^a > x^b$ 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.