F-vectors and h-vectors of simplicial posets

Abstract

A simplicial poset is a (finite) poset P with Ô such that every interval [Ô, x] is a boolean algebra. Simplicial posets are generalizations of simplicial complexes. The f-vector f( P) = ( f 0, f 1,…, f d −1) of a simplicial poset P of rank d is defined by f i =♯{ xε P: [Ô, x]≊ B i +1}, where B i +1 is a boolean algebra of rank i+1. We give a complete characterization of the f-vectors of simplicial posets and of Cohen-Macaulay simplicial posets, and an almost complete characterization for Gorenstein simplicial posets. The Cohen-Macaulay case relies on the theory of algebras with straightening laws (ASL's).