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B.2.2 General definitions for orderings

A monomial ordering (term ordering) on $K[x_1,\ldots,x_n]$ is a total ordering $<$ on the set of monomials (power products) $\{x^\alpha \mid \alpha \in \bf {N}^n\}$ which is compatible with the natural semigroup structure, i.e., $x^\alpha < x^\beta$ implies $x^\gamma
x^\alpha < x^\gamma x^\beta$ for any $\gamma \in \bf {N}^n$. 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 $M$ in $GL(n,R)$,but, of course, only integer coefficients are of relevance in practice.

Global orderings are wellorderings (i.e., $1 < x_i$ for each variable $x_i$), local orderings satisfy $1 > x_i$ 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 $K$ be the ground field, $x = (x_1, \ldots, x_n)$ the variables and $<$ a monomial ordering, then Loc $K[x]$ denotes the localization of $K[x]$ with respect to the multiplicatively closed set

\begin{displaymath}\{1 +
g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and }L(g) <

Here, $L(g)$ denotes the leading monomial of $g$, i.e., the biggest monomial of $g$ with respect to $<$. The result of any computation which uses standard basis computations has to be interpreted in Loc $K[x]$.

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.