The basis ingredience for all standard basis algorithms is the ordering of the monomials (and the concept of the leading term: the term with the highest monomial).

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
.

The ordering < is called a **wellordering** iff 1 is the smallest monomial.
Most of the algorithms work for general orderings.

Robbiano (cf.[R]) proved that any semigroup ordering can be defined
by a matrix
as follows (**matrix ordering**):

Let
be the rows of *A*, then
if and only if there is an *i* with
for *j* < *i* and
.
Thus,
if and only if
is smaller than
with
respect to the lexicographical ordering of vectors in
.

We call an ordering a **degree ordering** if it is given by a matrix with
coefficients of the first row either all positive or all negative.

Let *K* be a field; for
,
,
let
be the
**leading monomial** with respect to the ordering
<^{1}
and
the coefficient of
*L*(*g*) in *g*, that is
*g* = *c*(*g*)*L*(*g*) + smaller terms with respect to
<.

< is an **elimination ordering** for
iff
implies
).