| ### B.2.2 General definitions for orderings A monomial ordering (term ordering) on 62#62 is a total ordering 228#228 on the set of monomials (power products) 543#543 which is compatible with the natural semigroup structure, i.e., 544#544 implies 545#545 for any 546#546. We do not require 228#228 to be a wellordering. See the literature cited in References. It is known that any monomial ordering can be represented by a matrix 13#13 in 547#547,but, of course, only integer coefficients are of relevance in practice. Global orderings are wellorderings (i.e., 548#548 for each variable 126#126), local orderings satisfy 549#549 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 50#50 be the ground field, 550#550 the variables and 228#228 a monomial ordering, then Loc 551#551 denotes the localization of 551#551 with respect to the multiplicatively closed set 552#552 Here, 148#148 denotes the leading monomial of 149#149, i.e., the biggest monomial of 149#149 with respect to 228#228. The result of any computation which uses standard basis computations has to be interpreted in Loc 551#551. 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.

User manual for Singular version 4.4.1, 2025, generated by texi2html.