ABSTRACT
The Bourque–Ligh conjecture states that if is a gcd-closed set of positive integers with distinct elements, then the LCM matrix
is invertible. It is known that this conjecture holds for
but does not generally hold for
. 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
be a lattice, let
be a subset of P and let
be a function. We study under which conditions the join matrix
on S with respect to f is invertible on a meet closed set S (i.e.
.
Acknowledgements
We wish to thank Jori Mäntysalo for valuable help with Sage. We also like to thank the anonymous referee as well as the Editor, Professor Stéphane Gaubert for valuable comments that helped us to improve the paper.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Mika Mattila http://orcid.org/0000-0001-7946-6433