The spider poset is Macaulay

Let Q(k, l) be a poset whose Hasse diagram is a regular spider with k+ 1 le gs having the same length l. We show that for any n greater than or equal t o 1 the nth cartesian power of the spider poset Q(k, l) is a Macaulay poset for any k greater than or equal to 0 and l greater than or equal to 1. In combination with our recent results (S. L. Bezrukov, 1998, J. Combin. Theor y Ser. A 84, 157-170) this provides a complete characterization of all Maca ulay posers which are cartesian powers of upper semilattices, whose Hasse d iagrams are trees. (C) 2000 Academic Press.