## 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.

### MSC:

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

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