Determinants of matrices associated with incidence functions on posets. (English) Zbl 1080.11023

Summary: Let \(S=\{x_1,\dots ,x_n\}\) be a finite subset of a partially ordered set \(P\). Let \(f\) be an incidence function of \(P\). Let \([f(x_i\wedge x_j)]\) denote the \(n\times n\) matrix having \(f\) evaluated at the meet \(x_i\wedge x_j\) of \(x_i\) and \(x_j\) as its \(i,j\)-entry and \([f(x_i\vee x_j)]\) denote the \(n\times n\) 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 \(x_i\wedge x_j\in S\) for all \(1\leq i,j\leq n\). In this paper we get explicit combinatorial formulas for the determinants of matrices \([f(x_i\wedge x_j)]\) and \([f(x_i\vee x_j)]\) on any meet-closed set \(S\). We also obtain necessary and sufficient conditions for the matrices \([f(x_i\wedge x_j)]\) and \([f(x_i\vee x_j)]\) on any meet-closed set \(S\) to be nonsingular. Finally, we give some number-theoretic applications.


11C20 Matrices, determinants in number theory
15B57 Hermitian, skew-Hermitian, and related matrices
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