The Spider Poset Is Macaulay

Let Q(k, l) be a poset whose Hasse diagram is a regular spider with k+1 legs having the same length l. We show that for any n⩾1 the nth cartesian power of the spider poset Q(k, l) is a Macaulay poset for any k⩾0 and l⩾1. In combination with our recent results (S. L. Bezrukov, 1998, J. Combin. Theory Ser. A84, 157–170) this provides a complete characterization of all Macaulay posets which are cartesian powers of upper semilattices, whose Hasse diagrams are trees.