Monomial order

In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e.,

• If ${\displaystyle u\leq v}$ and ${\displaystyle w}$ is any other monomial, then ${\displaystyle uw\leq vw}$.

Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of being a Gröbner basis is always relative to a specific monomial order.