A monomial order in $k[x_1, x_2, \ldots, x_n]$ for a field $k$ is a relation $\prec$ on the monomials such that:
- $\prec$ is a total order;
- if $m_1 \prec m_2$ then $m_3m_1 \prec m_3m_2$ for any three monomials $m_1, m_2, m_3$; and
- any descending chain of monomials $m_1 \succ m_2 \succ \cdots$ terminates (is finite).
Notice that, if you knew only some of the comparisons in a given monomial order, you could determine some others. For example, given $x_1 \prec x_2$, we can determine that $x_1^2 \prec x_1x_2$. Therefore, to completely determine a monomial order, it suffices to give only a proper subset of the comparisons. What is the minimal set of comparisons that must be known before the whole monomial order is known? Can this set be finite?
Initially, I thought it might be enough to simply specify the order on the variables, i.e. the degree $1$ monomials. Evidently, this is not the case, since Wikipedia adopts the convention that $x_1 \succ x_2 \succ x_3 \succ \dots \succ x_n$ always. For example, given just those comparisons, I think that we cannot determine whether $x_1x_4 \prec x_2x_3$ or $x_1x_4 \succ x_2x_3$.