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Let $P$ be a finite poset. Let $L:=J(P)$ denote the lattice of order idealsof $P$. 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 $f_i(P)$ thenumber of the cliques $K_{i+1}$, contained in the graph $G(P)$. Our mainresults are some linear equations connecting the numbers $f_i(P)$ and $b_i(L)$.We reprove the Dehn--Sommerville equations for simplicial polytopes. In ourproof we use free resolutions and the theory of Stanley--Reisner rings.

Linear equations for the number of intervals which are isomorphic with Boolean lattices and the Dehn--Sommerville equations