Some further applications of a lattice theoretic method in the study of singular LCM matrices

TL;DR

This paper explores the singularity of GCD-closed LCM matrices using lattice theory, extending previous results to sets of size nine and providing solutions to open questions, along with a new proof of Sun's conjecture.

Contribution

It applies lattice theoretic methods to analyze singular LCM matrices for nine-element sets and addresses open questions from prior research, including a new proof of Sun's conjecture.

Findings

Characterization of lattice structures leading to singular matrices for n=9

Solutions to open questions about lattice configurations

A new lattice theoretic proof of Sun's conjecture

Abstract

In 1876 H. J. S. Smith defined an LCM matrix as follows: let S = {x_1, x_2, ..., x_n} be a set of positive integers. The LCM matrix [S] is the n $\times$ n matrix with lcm(x_i , x_j) as its ij entry. During the last 30 years singularity of LCM matrices has interested many authors. In 1992 Bourque and Ligh ended up conjecturing that if the GCD closedness of the set S (which means that gcd(x_i, x_j) $\in$ S for all i, j $\in$ {1, 2, . . . , n}), suffices to guarantee the invertibility of the matrix [S]. However, a few years later this conjecture was proven false first by Haukkanen et al. and then by Hong. It turned out that the conjecture holds only on GCD closed sets with at most 7 elements but not in general for larger sets. However, the given counterexamples did not give much insight on why does the conjecture fail exactly in the case when n=8. This situation was later improved in a couple of articles, where a new lattice theoretic approach was introduced (the method is based on the fact that because the set S is assumed to be GCD closed, the structure (S, |) actually forms a meet semilattice). For example, it has been shown that in the case when the set S has 8 elements and the matrix [S] is singular, there is only one option for the semilattice structure of (S, |), namely the cube structure. Since the cases up to n=8 have been thoroughly studied in various articles, the next natural step is to apply the methods to the case n=9. This was done by Altinisik and Altintaa as they consider the different lattice structures of (S, |) with nine elements that can result in a singular LCM matrix [S]. However, their investigation leaves two open questions, and the main purpose of this presentation is to provide solutions to them. We shall also give a new lattice theoretic proof for a result referred to as Sun's conjecture, which was originally proven by Hong via number theoretic approach.