Studying the inertias of LCM matrices and revisiting the Bourque-Ligh conjecture

TL;DR

This paper investigates the inertia of LCM matrices of GCD closed sets, introduces a new lattice-theoretic concept to analyze their properties, and explores conditions under which the Bourque-Ligh conjecture holds or fails.

Contribution

It introduces the concept of double-chain sets in lattices, analyzes the inertia of LCM matrices using lattice structures, and provides methods to construct matrices with predominantly positive or negative eigenvalues.

Findings

The inertia of LCM matrices can often be determined by lattice structure.

The Bourque-Ligh conjecture holds in many cases but fails in specific instances.

Methods are provided to construct LCM matrices with mostly positive or negative eigenvalues.

Abstract

Let $S=\{x_1,x_2,\ldots,x_n\}$ be a finite set of distinct positive integers. Throughout this article we assume that the set $S$ is GCD closed. The LCM matrix $[S]$ of the set $S$ is defined to be the $n\times n$ matrix with $\mathrm{lcm}(x_i,x_j)$ as its $ij$ element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set $S$ is always invertible, but currently it is a well-known fact that any nontrivial LCM matrix is indefinite and under the right circumstances it can be even singular (even if the set $S$ is assumed to be GCD closed). However, not much more is known about the inertia of LCM matrices in general. The ultimate goal of this article is to improve this situation. Assuming that $S$ is a meet closed set we define an entirely new lattice-theoretic concept by saying that an element $x_i\in S$ generates a double-chain set in $S$ if the set $\mathrm{meetcl}(C_S(x_i))\setminus C_S(x_i)$ can be expressed as a union of two disjoint chains (here the set $C_S(x_i)$ consists of all the elements of the set $S$ that are covered by $x_i$ and $\mathrm{meetcl}(C_S(x_i))$ is the smallest meet closed subset of $S$ that contains the set $C_S(x_i)$). We then proceed by studying the values of the M\"obius function on sets in which every element generates a double-chain set and use the properties of the M\"obius function to explain why the Bourque-Ligh conjecture holds in so many cases and fails in certain very specific instances. After that we turn our attention to the inertia and see that in some cases it is possible to determine the inertia of an LCM matrix simply by looking at the lattice-theoretic structure of $(S,|)$ alone. Finally, we are going to show how to construct LCM matrices in which the majority of the eigenvalues is either negative or positive.