[Math] Is reverse lexicographic order the same as graded reverse lexicographic order

Question

I want to make sure whether the two monomial orderings are actually the same thing. I am confused because the Cox book on Ideals, Varieties and Algorithms mentions only the graded reverse lexicographic order.

My guess is that they are two different monomial orderings because Macaulay2 at least has two separate commands for the graded reverse lexicographic order and the reverse lexicorgraphic order here so I think they must be different things despite the book having only the grevlex, not revlex.

So

Is reverse lexicographic order (revlex) the same thing as the graded reverse lexicographic order (grevlex)?

Answer 1

Yes they are different. Macaulay2 defines them this way:

1. "The graded reverse lexicographic order is defined by: $x^A > x^B$ if either $degree(x^A) > degree(x^B)$ or $degree(x^A) = degree(x^B)$ and the LAST non-zero entry of the vector of integers $A-B$ is NEGATIVE."

2. "The reverse lexicographic order is defined by: $x^A > x^B$ if the FIRST non-zero entry of the vector of integers $A-B$ is NEGATIVE. This is a local order, not a global order. Therefore Gröbner bases over this ring only give generators over the local ring whose fractions are all elements not in the ideal generated by the variables."