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.