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B.2.2 General definitions for orderings

A monomial ordering (term ordering) on K[x1,,xn] is a total ordering < on the set of monomials (power products) {xαα Nn} which is compatible with the natural semigroup structure, i.e., xα < xβ implies xγxα < xγxβ for any γ Nn. We do not require < to be a well ordering.

See the literature cited in References.

It is known that any monomial ordering can be represented by a matrix M in GL(n,R), but, of course, only integer coefficients are of relevance in practice.

Global orderings are well orderings (i.e., 1 < xi for each variable xi), local orderings satisfy 1 > xi for each variable. If some variables are ordered globally and others locally we call it a mixed ordering. Local or mixed orderings are not well orderings.

Let K be the ground field, x = (x1,,xn) the variables and < a monomial ordering, then Loc K[x] denotes the localization of K[x] with respect to the multiplicatively closed set

{1 + g | g = 0 or g ∈ K [x]\{0} and L(g) < 1}.
Here, L(g) denotes the leading monomial of g, i.e., the biggest monomial of g with respect to <. The result of any computation which uses standard basis computations has to be interpreted in Loc K[x].

Note that the definition of a ring includes the definition of its monomial ordering (see Rings and orderings). SINGULAR offers the monomial orderings described in the following sections.


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