- 1.
- Computations of the degree (resp. weighted degree):

the*degree*(resp.*weighted degree*) of a monomial is the sum of the exponents (resp. the weighted sum with respect to a weight vector*w*: ). - 2.
- Test for divisibility:

. - 3.
- Addition of two monomials:

. - 4.
- Comparison of two monomials with respect to a monomial ordering.

A *monomial ordering* > (term ordering) on the set of monomials
*M*_{n} is a total ordering on *M*_{n} which is compatible with the
natural semigroup structure, i.e.,
implies
for any
.
A monomial ordering
is a well-ordering if
is the smallest
monomial. We furthermore call an ordering negative if
is the largest monomial.

Robbiano (cf.[15]) proved that any monomial ordering > can be defined by a matrix : . Matrix-based descriptions of monomial orderings are very general, but have the disadvantage that their realization in an actual implementation is usually rather time-consuming. Therefore, they are not very widely used in practice.

Instead, the most frequently used descriptions of orderings
have at most two defining conditions: a (possibly weighted) degree and
a (normal
or reverse) lexicographical comparison. We call such orderings
*simple orderings*. The most important simple orderings (and
their SINGULAR abbreviations) are:

- lexicographical (lp): used to eliminate variables and solve equations,
- (weighted) degree reverse lexicographical (dp): in general the most efficient one to compute a GB (as shown in [3]),
- (weighted) degree lexicographical (Dp),
- negative lexicographical ordering (ls),
- (weighted) negative degree reverse lexicographical (ds): also called tangent cone ordering,
- (weighted) negative degree lexicographical (Ds).

For monomials
let

Then we can define for the above mentioned simple monomial orderings by:

We furthermore call a monomial ordering > a *degree based
monomial ordering* if
(e.g., Dp and
dp and their weighted relatives are degree based orderings).

Due to the nature of the GB algorithm, monomial operations are by far the most frequently used primitive operations. For example, monomial comparisons are performed much more often than, and monomial additions at least as often as, arithmetic operations over the coefficient field. The number of divisibility tests depends very much on the given input ideal, but is usually very large, as well (see also Table 1).

Nevertheless, whether or not monomial operations dominate the running
time of a GB computation depends on the coefficient field of the
underlying polynomial ring: monomial operations are certainly
run-time dominating for finite fields with a small^{} characteristic (e.g., integers modulo a small prime
number), since an arithmetic operation over these fields can usually
be realized much faster than a monomial operation. However, for fields
of characteristic 0 (like the rational numbers), GB computations are
usually dominated by the arithmetic operations over these fields,
since the time needed for these operations is proportional to the size
of the coefficients which tend to grow rapidly during a GB
computation.

Therefore, improvements in the efficiency of monomial operations will have less of an impact on GB computations over fields of characteristic 0.