A Local-Global Principle for Macaulay Posets

We consider the shadow minimization problem (SMP) for cartesian powers Pn of a Macaulay poset P . Our main result is a local-global principle with respect to the lexicographic order Ln. Namely, we show that under certain conditions the shadow of any initial segment of the order Ln for n ≥ 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.