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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 $\times$ nmatrix with lcm(x_i , x_j) as its ij entry. During the last 30 yearssingularity of LCM matrices has interested many authors. In 1992 Bourque andLigh ended up conjecturing that if the GCD closedness of the set S (which meansthat gcd(x_i, x_j) $\in$ S for all i, j $\in$ {1, 2, . . . , n}), suffices toguarantee the invertibility of the matrix [S]. However, a few years later thisconjecture was proven false first by Haukkanen et al. and then by Hong. Itturned out that the conjecture holds only on GCD closed sets with at most 7elements but not in general for larger sets. However, the given counterexamplesdid not give much insight on why does the conjecture fail exactly in the casewhen n=8. This situation was later improved in a couple of articles, where anew lattice theoretic approach was introduced (the method is based on the factthat 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 thecase when the set S has 8 elements and the matrix [S] is singular, there isonly one option for the semilattice structure of (S, |), namely the cubestructure.  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 byAltinisik 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 thispresentation is to provide solutions to them. We shall also give a new latticetheoretic proof for a result referred to as Sun's conjecture, which wasoriginally proven by Hong via number theoretic approach.

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