Reading this paper I stumbled upon a Groebner Basis calculation with a custom monomial order prescription. The algorithm 2) which you can find on page 11 of the above paper deals with polynomials in variables $w$ and $z_1,z_2,...,z_n$. It suggests the following monomial order in these variables:
1) First compare the degrees of $w$.
2) Then apply grevlex order for $z_1,z_2,...,z_n$, so that $z_1\succ...\succ z_n$.
I have two questions concerning the above. First of all, as far as I understand explicitly writing $z_1\succ...\succ z_n$ means that any power of $z_i$ has higher weight than any power of $z_j$ if $i<j$, and then the powers of $z_i$ among themselves get ordered as usual (higher power means higher weight). That is basically what grevlex means. Is this correct? And the second question is: why do the authors separate the ordering conventions into two steps 1) and 2) instead of just writing $w\succ z_1\succ...\succ z_n$? Doesn't $w\succ z_1\succ...\succ z_n$ imply exactly the same ordering as the two step ordering given by the authors? Thanks for any suggestion.
You first compare the degrees in $w$. If these are equal, you compare the total degrees in the remaining indeterminates. For the same total degree in these indeterminates, you use the reverse lexicographic order (i. e. you first look at the degree in $z_n$; if the degrees are different, the smallest degree yields the greatest monomial).