A lattice-theoretic approach to the Bourque-Ligh conjecture

The Bourque-Ligh conjecture states that if $S=\{x_1,x_2,\ldots,x_n\}$ is a gcd-closed set of positive integers with distinct elements, then the LCM matrix $[S]=[\hbox{lcm}(x_i,x_j)]$ is invertible. It is well known that this conjecture holds for $n\leq7$ but does not generally hold for $n\geq8$. In this paper we provide a lattice-theoretic explanation for this solution of the Bourque-Ligh conjecture. In fact, let $(P,\leq)=(P,\land,\lor)$ be a lattice, let $S=\{x_1,x_2,\ldots,x_n\}$ be a subset of $P$ and let $f:P\to{\mathbb C}$ be a function. We study under which conditions the join matrix $[S]_f=[f(x_i\lor x_j)]$ on $S$ with respect to $f$ is invertible on a meet closed set $S$ (i.e., $x_i,x_j\in S\Rightarrow x_i\land x_j\in S)$.