Mathematical College, Sichuan University, Chengdu 610064, P.R. China, e-mails: s-f.hong@tom.com, hongsf02@yahoo.com
Abstract: 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 $1\le i,j\le n$. 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.
Keywords: meet-closed set, greatest-type lower, incidence function, determinant, nonsingularity
Classification (MSC 2000): 11C20, 15A57
Full text available as PDF (smallest), as compressed PostScript (.ps.gz) or as raw PostScript (.ps).
Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade.
To activate your access, please contact Myris Trade at myris@myris.cz.
Subscribers of Springer (formerly Kluwer) need to access the articles on their site, which is http://www.springeronline.com/10587.
