# Article

 Title: Determinants of matrices associated with incidence functions on posets (English) Author: Hong, Shaofang Author: Sun, Qi Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 54 Issue: 2 Year: 2004 Pages: 431-443 Summary lang: English . Category: math . Summary: Let $S=\lbrace x_1,\dots ,x_n\rbrace$ 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 $1\le i,j\le n$. 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. (English) Keyword: meet-closed set Keyword: greatest-type lower Keyword: incidence function Keyword: determinant Keyword: nonsingularity MSC: 06A07 MSC: 06A12 MSC: 11C20 MSC: 15A57 idZBL: Zbl 1080.11023 idMR: MR2059264 . Date available: 2009-09-24T11:14:14Z Last updated: 2020-07-03 Stable URL: http://hdl.handle.net/10338.dmlcz/127901 . Reference: [1] M.  Aigner: Combinatorial Theory.Springer-Verlag, New York, 1979. Zbl 0415.05001, MR 0542445 Reference: [2] S.  Beslin, S.  Ligh: Greatest common divisor matrices.Linear Algebra Appl. 118 (1989), 69–76. MR 0995366 Reference: [3] S.  Beslin, S.  Ligh: Another generalization of Smith’s determinant.Bull. Austral. Math. Soc. 40 (1989), 413–415. MR 1037636, 10.1017/S0004972700017457 Reference: [4] K.  Bourque, S.  Ligh: Matrices associated with arithmetical functions.Linear and Multilinear Algebra 34 (1993), 261–267. MR 1304611, 10.1080/03081089308818225 Reference: [5] P.  Haukkanen: On meet matrices on posets.Linear Algebra Appl. 249 (1996), 111–123. Zbl 0870.15016, MR 1417412, 10.1016/0024-3795(95)00349-5 Reference: [6] S.  Hong: LCM matrix on an $r$-fold gcd-closed set.J.  Sichuan Univ., Nat. Sci. Ed. 33 (1996), 650–657. Zbl 0869.11021, MR 1440627 Reference: [7] S.  Hong: On the Bourque-Ligh conjecture of least common multiple matrices.J.  Algebra 218 (1999), 216–228. Zbl 1015.11007, MR 1704684, 10.1006/jabr.1998.7844 Reference: [8] S.  Hong: On the factorization of LCM matrices on gcd-closed sets.Linear Algebra Appl. 345 (2002), 225–233. Zbl 0995.15006, MR 1883274 Reference: [9] D.  Rearick: Semi-multiplicative functions.Duke Math.  J. 33 (1966), 49–53. Zbl 0154.29503, MR 0184897, 10.1215/S0012-7094-66-03308-4 Reference: [10] H. J. S.  Smith: On the value of a certain arithmetical determinant.Proc. London Math. Soc. 7 (1875–1876), 208–212. .

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