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B.2.2 General definitions for orderingsA monomial ordering (term ordering) on K[x1,…,xn] is a total ordering < on the set of monomials (power products) {xα∣α ∈ 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 ![]() 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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