Singular Manual: General definitions for orderings

| ### 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.

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