Invertible relations on polytopes (Q1093907)

Let A(P) be the incidence algebra defined on the lattice (the ordering is defined by set inclusion) of faces of a convex polytope P. (Remember that 1) \(f\in A(P)\) is a function defined on pairs of faces of P with \(f(F^ i,F^ j)=0\) unless \(F^ i\subset F^ j\), and 2) for f,g\(\in A(P)\), the function \(f\circ g\in A(P)\) is defined by \((f\circ g)(F^ i,F^ j)=\sum \{f(F^ i,F^ k)g(F^ k,F^ j): F^ i\subset F^ k\subset F^ j\}\); here \(F^ i\) denotes an i-dimensional face of P.) Theorem 1. Let \(f\in A(P)\) be such that \(f(F^ i,F^ j)=f(i,j)=G(i)/H(j)\) where G and H are nonzero functions on integers. Then \(f^{-1}(F^ i,F^ j)=g(i,j)=(-1)^{j-i}H(i)/G(j).\) Theorem 2. Let \(f\in A(P)\) be such that \(f(F^ i,F^ j)=f(i,j)\neq 0\) and \(f^{-1}(F^ i,F^ j)=g(i,j)\). Then there exist nonzero functions G and H on the integers and a constant c such that \(f(i,i)=1/g(i,i)=G(i)/H(i)\) and, for \(i<j\), \(f(i,j)=G(i)/H(j)z(j-1)\), \(g(i,j)=(-1)^{j-i}H(i)/G(j)z(i)\), where \(z(k)=1+c\) if k is even and \(z(k)=1-c\) if k is odd.