A lattice-theoretic approach to the Bourque?Ligh conjecture

The Bourque-Ligh conjecture states that if S = {x1, x2,..., xn} is a gcd-closed set of positive integers with distinct elements, then the LCM matrix [S] = [lcm(xi, xj)] is invertible. It is known that this conjecture holds for n = 7 but does not generally hold for n = 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,=) = (P,.,.) be a lattice, let S = {x1, x2,..., xn} be a subset of P and let f : P. C be a function. We study under which conditions the join matrix [S] f = [f (xi. xj)] onS with respect to f is invertible on a meet closed set S (i.e. xi, xj. S. x(i) boolean AND x(j) is an element of S).