In particular, the property of being a Gröbner basis is always relative to a specific monomial order.
Besides respecting multiplication, monomial orders are often required to be well-orders, since this ensures the multivariate division procedure will terminate.
There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders.
In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions: Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order.
The choice of a total order on the monomials allows sorting the terms of a polynomial.
Then the set M of the (monic) monomials in R is a basis of R, considered as a vector space over the field of the coefficients.
as a linear combination of monomials, where S is a finite subset of M and the cu are all nonzero.
In this article, a monomial is assumed to not include a coefficient.
The defining property of monomial orderings implies that the order of the terms is kept when multiplying a polynomial by a monomial.
Therefore, the notion of monomial order becomes interesting only in the case of multiple variables.
With this convention there are still many examples of different monomial orders.
Lexicographic order (lex) first compares exponents of x1 in the monomials, and in case of equality compares exponents of x2, and so forth.
The name is derived from the similarity with the usual alphabetical order used in lexicography for dictionaries, if monomials are represented by the sequence of the exponents of the indeterminates.
If the number of indeterminates is fixed (as it is usually the case), the lexicographical order is a well-order, although this is not the case for the lexicographical order applied to sequences of various lengths.
Graded lexicographic order (grlex, or deglex for degree lexicographic order) first compares the total degree (sum of all exponents), and in case of a tie applies lexicographic order.
Although very natural, this ordering is rarely used: the Gröbner basis for the graded reverse lexicographic order, which follows, is easier to compute and provides the same information on the input set of polynomials.
Graded reverse lexicographic order (grevlex, or degrevlex for degree reverse lexicographic order) compares the total degree first, then uses a lexicographic order as tie-breaker, but it reverses the outcome of the lexicographic comparison so that lexicographically larger monomials of the same degree are considered to be degrevlex smaller.
For the final order to exhibit the conventional ordering x1 > x2 > ... > xn of the indeterminates, it is furthermore necessary that the tie-breaker lexicographic order before reversal considers the last indeterminate xn to be the largest, which means it must start with that indeterminate.
A concrete recipe for the graded reverse lexicographic order is thus to compare by the total degree first, then compare exponents of the last indeterminate xn but reversing the outcome (so the monomial with smaller exponent is larger in the ordering), followed (as always only in case of a tie) by a similar comparison of xn−1, and so forth ending with x1.
The differences between graded lexicographic and graded reverse lexicographic orders are subtle, since they in fact coincide for 1 and 2 indeterminates.
The first difference comes for degree 2 monomials in 3 indeterminates, which are graded lexicographic ordered as
The general trend is that the reverse order exhibits all variables among the small monomials of any given degree, whereas with the non-reverse order the intervals of smallest monomials of any given degree will only be formed from the smallest variables.
Block order or elimination order (lexdeg) may be defined for any number of blocks but, for sake of simplicity, we consider only the case of two blocks (however, if the number of blocks equals the number of variables, this order is simply the lexicographic order).
This ordering is important as it allows elimination, an operation which corresponds to projection in algebraic geometry.
It first compares the dot product of the exponent sequences of the monomials with this weight vector, and in case of a tie uses some other fixed monomial order.
If the ai are rationally independent numbers (so in particular none of them are zero and all fractions
are irrational) then a tie can never occur, and the weight vector itself specifies a monomial ordering.
In the contrary case, one could use another weight vector to break ties, and so on; after using n linearly independent weight vectors, there cannot be any remaining ties.
One can in fact define any monomial ordering by a sequence of weight vectors (Cox et al. pp.
For example, graded reverse lexicographic order has a reputation for producing, almost always, the Gröbner bases that are the easiest to compute (this is enforced by the fact that, under rather common conditions on the ideal, the polynomials in the Gröbner basis have a degree that is at most exponential in the number of variables; no such complexity result exists for any other ordering).