The most important monomial order is the (graded) reverse lexicographic order. This is the default in Macaulay, since it can be shown that, in many cases, it is theoretically the most efficient [BS1].
Thus, in k[x,y,z],
There are many reverse lexicographic orders, in fact, there is one for each permutation of the variables.
The user can define the reverse lexicographic order using the ring command. For example, to define a polynomial ring with five variables of weight one, e.g. (a, b, c, d, e) , and using the reverse lexicographic order in this ring, you need to simply input the number of variables and <return> (the default) for the monomial order.
% ring R ! characteristic (if not 31991) ? <return> ! number of variables ? 5 ! 5 variables, please ? abcde ! variable weights (if not all 1) ? <return> ! monomial order (if not rev. lex.) ? <return> ; largest degree of a monomial : 217
As this example illustrates, you don't have to do anything special to
get the reverse lexicographic order. Note that this order is
graded: it first orders by degree. For example, in k[x,y,z], if
, 
, and 
, then the order of monomials
in the reverse lexicographic order is
since
.