On Posets whose Products are Macaulay

If P is an upper semilattice whose Hasse diagram is a tree and whose cartesian powers are Macaulay, it is shown that Hasse diagram of P is actually a spider in which all the legs have the same length. 1 Introduction Let (P; `) be a finite poset. For x; y 2 P we write x ae \Delta y if x ae y and there is no z 2 P yielding x ae z ae y. The poset (P; `) is called ranked if there exists a function r P : P 7! IN such that min x2P r P (x) = 0 and for any x; y 2 P the condition x ae\Delta y implies r P (x) + 1 = r P (y). We call the numbers r P (x) and r P = max x2P r P (x) the rank of x and P respectively. The set P i = fx 2 P j r P (x) = ig is called the i th level of P . For a subset A ` P i and i ? 0 define the shadow of A as \Delta(A) = fx 2 P i\Gamma1 j x ae\Delta y for some y 2 Ag: The the shadow minimization problem plays an important role in combinatorics and is often in the background of various extremal problems: for a given poset (P; `) and given natural numbers i ? 0..