×

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
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] M. Aigner: Combinatorial Theory. Springer-Verlag, New York, 1979. · Zbl 0415.05001
[2] S. Beslin, S. Ligh: Greatest common divisor matrices. Linear Algebra Appl. 118 (1989), 69-76. · Zbl 0672.15005
[3] S. Beslin, S. Ligh: Another generalization of Smith’s determinant. Bull. Austral. Math. Soc. 40 (1989), 413-415. · Zbl 0675.10002
[4] K. Bourque, S. Ligh: Matrices associated with arithmetical functions. Linear and Multilinear Algebra 34 (1993), 261-267. · Zbl 0815.15022
[5] P. Haukkanen: On meet matrices on posets. Linear Algebra Appl. 249 (1996), 111-123. · Zbl 0870.15016
[6] S. Hong: LCM matrix on an \(r\)-fold gcd-closed set. J. Sichuan Univ., Nat. Sci. 33 (1996), 650-657. · Zbl 0869.11021
[7] S. Hong: On the Bourque-Ligh conjecture of least common multiple matrices. J. Algebra 218 (1999), 216-228. · Zbl 1015.11007
[8] S. Hong: On the factorization of LCM matrices on gcd-closed sets. Linear Algebra Appl. 345 (2002), 225-233. · Zbl 0995.15006
[9] D. Rearick: Semi-multiplicative functions. Duke Math. J. 33 (1966), 49-53. · Zbl 0154.29503
[10] H. J. S. Smith: On the value of a certain arithmetical determinant. Proc. London Math. Soc. 7 (1875-1876), 208-212.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.