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

In 1876 H. J. S. Smith defined an LCM matrix as follows: let S={x1,x2,…,xn} be a set of positive integers with x1<x2<⋯<xn. The [S] on the S is n×n lcm(xi,xj) its ij entry. During last 30 years singularity matrices has interested many authors. 1992 Bourque and Ligh ended up conjecturing that if GCD closedness (which means gcd⁡(xi,xj)∈S for all i,j∈{1,2,…,n}), suffices to guarantee invertibility [S]. However, few later this conjecture was proven false first by Haukkanen et al. then Hong. It turned out holds only closed sets at most 7 elements but not in general larger sets. given counterexamples did give much insight why does fail exactly case when n=8. This situation improved couple articles, where new lattice theoretic approach introduced (the method based fact because assumed closed, structure (S,|) actually forms meet semilattice). For example, it been shown 8 singular, there one option semilattice (S,|), namely cube structure. Since cases n≤8 have thoroughly studied various next natural step apply methods n=9. done Altınışık Altıntaş they consider different structures nine can result singular their investigation leaves two open questions, main purpose presentation provide solutions them. We shall also proof referred Sun's conjecture, which originally Hong via number approach.