Determinants of Matrices Associated with Incidence Functions on Posets

Abstract

Let S = x 1,...,x n} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let $$\left[ {f\left( {x_i \Lambda x_j } \right)} \right]$$ denote the n × n matrix having f evaluated at the meet $${x_i \Lambda x_j }$$ of x i and x j as its i, j-entry and $$\left[ {f\left( {x_i \vee x_j } \right)} \right]$$ denote the n × n matrix having f evaluated at the join $$x_i \vee x_j $$ of x i and x j as its i, j-entry. The set S is said to be meet-closed if $$\left[ {f\left( {x_i \vee x_j } \right)} \right]$$ for all 1 ≤ i, j ≤ n. In this paper we get explicit combinatorial formulas for the determinants of matrices $$\left[ {f\left( {x_i \Lambda x_j } \right)} \right]$$ and $$\left[ {f\left( {x_i \vee x_j } \right)} \right]$$ on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices $$\left[ {f\left( {x_i \Lambda x_j } \right)} \right]$$ and $$\left[ {f\left( {x_i \vee x_j } \right)} \right]$$ on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications.