The monomial order which Macaulay uses is the strict lexicographic order on the 32 bit integers which form the internal representation of a monomial. For example,
% ring R ! characteristic (if not 31991) ? <return> ! number of variables ? 7 ! 7 variables, please ? xyza-d ! variable weights (if not all 1) ? 1:3 4 ! monomial order (if not rev. lex.) ? w 3 4 ! weight vector #1 ? 1 0:6 ; largest degree of a monomial : 512 512 % pring R ; ring R ; characteristic : 31991 ; number of variables : 7 ; 3 variables for block 1 : xyz ; 4 variables for block 2 : abcd ; weights for block 1 : 1 1 1 ; weights for block 2 : 4 4 4 4 ; weight vector block 0 : 1 0 0 0 0 0 0 ; monomial order : w 3 4 c ; top degree of a monomial : 512 512 % poly f x2+xy10a+xy10+a50+a53b % type f ; x2+xy10a+xy10+a53b+a50 % ; The weight vector (1,0,...,0) takes precedence: the monomials % ; are ordered first by highest power in "x", then by the (graded) % ; reverse lexicographic order in x,y,z (ignoring a,...,d), and finally % ; by the reverse lex. order in a,b,c,d.