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B.2.2 General definitions for orderingsA monomial ordering (term ordering) on is a total ordering on the set of monomials (power products) which is compatible with the natural semigroup structure, i.e., implies for any . 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 in , but, of course, only integer coefficients are of relevance in practice. Global orderings are well orderings (i.e., for each variable ), local orderings satisfy 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 be the ground field,
the
variables and a monomial ordering, then Loc denotes the
localization of with respect to the multiplicatively closed set Here, denotes the leading monomial of , i.e., the biggest monomial of with respect to . The result of any computation which uses standard basis computations has to be interpreted in Loc . 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. |