B.2.2 General definitions for orderings
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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 well ordering.
 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 well orderings (i.e.,  \hbox{$1 < x_i$} 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 well orderings.

Let $K$ be the ground field, \hbox{$x = (x_1, \ldots, x_n)$} the
variables and $<$ a monomial ordering, then Loc $K[x]$ denotes the
localization of $K[x]$ with respect to the multiplicatively closed set $$\{1 +
g \mid g = 0 \hbox{ or } g \in K[x]\backslash \{0\} \hbox{ and }L(g) <
1\}.$$   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. 


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