  Let S be a polynomial ring in n variables over a field, and let M be a
monomial ideal of S. We introduce a new invariant, called the order of
dominance of S/M, denoted odom(S/M), which has many similarities with the
codimension of S/M. We use this order of dominance to characterize the class of
Scarf ideals that are Cohen-Macaulay, and also to characterize when the Taylor
resolution is minimal. We also show that odom(S/M) has the following
properties: (i) codim(S/M) <= odom(S/M) <= pd(S/M). (ii) pd(S/M)=n if and only
if odom(S/M)=n. (iii) pd(S/M)=1 if and only if odom(S/M)=1. (iv) If
odom(S/M)=n-1 then pd(S/M)=n-1.
