Monomial order

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In mathematics, a monomial order is a total order on the set of all monomials (considering monomials which only differ in their coefficient to be the same) satisfying two additional properties.

1. If u < v and w is any other monomial, then uw<vw. In other words, the ordering respects multiplication.
2. The ordering is a well ordering

One example of a monomial ordering is the lexicographic order. Another is ordering monomials by total degree and then using lexicographic order to order polynomials of the same degree.

More generally you can also allow orderings which do not satisfy condition 2. Orderings satisfying 2 are called global orderings. Being a global ordering is equivalent, for polynomial rings in finitely many variable,s to the property that all variables x_i are greater than 1.

Orderings with the contrary property, that all variables x_i are smaller than 1, are called local orderings. Orderings which are neither global nor local are called mixed orderings.