LINEAR EQUATIONS FOR THE NUMBER OF INTERVALS WHICH ARE ISOMORPHIC WITH BOOLEAN LATTICES AND THE DEHN-SOMMERVILLE EQUATIONS

Let P be a finite poset. Let L := J(P) denote the lattice of order ideals of P. Let b(i)(L) denote the number of Boolean intervals of L of rank i. We construct a simple graph G(P) from our poset P. Denote by f(i)(P) the number of the cliques Ki+1, contained in the graph G(P). Our main results are some linear equations connecting the numbers f(i)(P) and b(i)(L). We reprove the Dehn-Sommerville equations for simplicial polytopes. In our proof, we use free resolutions and the theory of Stanley-Reisner rings.