is the set of linear combinations of elements of with coefficients in ; that is the set of polynomials that can be written with

Monomial ordering

All operations related to Gröbner bases require the choice of a total order on the monomials, with the following properties of compatibility with multiplication. For all monomials , , ,
#
# .A total order satisfying these condition is sometimes called an admissible ordering.

These conditions imply that the order is a well-order,

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