A lattice-theoretic approach to the Bourque–Ligh conjecture

ABSTRACT

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 $\left[S\right]=\left[\text{lcm}\right(x_i,x_j\left)\right]$ is invertible. It is known that this conjecture holds for $n\le 7$ but does not generally hold for $n\ge 8$. 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 $(P,\le )=(P,\wedge ,\vee )$ 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. $x_i,x_j\in S\Rightarrow x_i\wedge x_j\in S)$.

KEYWORDS:

• Bourque–Ligh conjecture
• join matrix
• LCM matrix
• GCD matrix
• semimultiplicative function

AMS SUBJECT CLASSIFICATIONS:

• 11C20
• 15B36
• 06A07

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