A local-global principle for Macaulay posets

We consider the shadow minimization problem (SMP) for Cartesian powers P-n of a Macaulay poset P. Our main result is a local-global principle with respect to the lexicographic order L-n. Namely, we show that under certain conditions the shadow of any initial segment of the order L-n for n greater than or equal to 3 is minimal iff it is so for n = 2. These conditions include such poset properties as additivity, shadow increasing, final shadow increasing and being rank-greedy. We also show that these conditions are essentially necessary for the lexicographic order to provide nestedness in the SMP.