A monomial order is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication.

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 $ u \le v $ and $ w $ is any other monomial, then $ uw \le 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.

Besides respecting multiplication, monomial orders are often required to be well-orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders.

In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions:

  1. The order is a total order.
  2. If $ u $ is any monomial then $ 1 \le u $.

Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order.

Source: Wikipedia