B.2.2 General definitions for orderings
A 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 wellordering. 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 wellorderings (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 wellorderings.
Let be the ground field,
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.