¹‘≪’ are called multigraded lexicographic orders. Intuitively, we think of A, B and C as corresponding to variables in some engineering problem which represent quantities which are known and think of a, b, c, d, e and f as corresponding to variables in the engineering problem which represent quantities which are unknown. The fact that d, e and f are in the top level indicates that we are very interested in solving for d, e and f in terms of A, B, C, a, b and c, but are not willing to solve for b in terms of expressions involving either d, e or f.

More precise discussion of mulitgraded lex orderings are in Chapter 18.