Summary: The Bourque-Ligh conjecture states that if \(S=\{x_1,x_2,\dots,x_n\}\) is a gcd-closed set of positive integers with distinct elements, then the LCM matrix \([S] =[\mathrm{lcm}(x_i,x_j)]\) is invertible. It is known that this conjecture holds for \(n\leq 7\) but does not generally hold for \(n\geq 8\). In this paper, we study the invertibility of matrices in a more general matrix class, join matrices. At the same time, we provide a lattice-theoretic explanation for this solution of the Bourque-Ligh conjecture. In fact, let \((P,\leq)=(P\wedge,\vee)\) be a lattice, let \(S=\{x_1,x_2,\dots, x_n\}\) be a subset of \(P\) and let \(f:P\to\mathbb{C}\) be a function. We study under which conditions the join matrix \([S]_f=[f(x_i\vee x_j)]\) on \(S\) with respect to \(f\) is invertible on a meet closed set \(S\) (i.e. \(x_i\), \(x_i\in S\Rightarrow x_i\wedge x_j\in S)\).