Monomial Order

In mathematics, a monomial order is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the following two properties:

1. If u < v and w is any other monomial, then uw. In other words, the ordering respects multiplication.
2. The ordering is a well ordering (every non-empty set of monomials has a minimal element).

Among the powers of any one variable x, the only ordering satisfying these conditions is the natural ordering 1<xx would fail to have a minimal element). Therefore the notion of monomial ordering is interesting only in the case of multiple variables.

Monomial orderings are most commonly used with Gröbner bases and multivariate division.