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The invertibility of LCM matrices and their Hadamard powers have been studieda lot over the years by many authors. Bourque and Ligh conjectured in 1992 thatthe LCM matrix $[S]=[[x_i, x_j]]$ on any GCD closed set $S=\{x_1, x_2, \ldots,x_n\}$ is invertible, but in 1997 this was proven false. However, currentlythere are many open conjectures concerning LCM matrices and their Hadamardpowers presented by Hong. In this paper we utilize lattice-theoretic structuresand the M\"obius function to explain the singularity of classical LCM matricesand their Hadamard powers. At the same time we end up disproving some of Hong'sconjectures. Elementary mathematical analysis is applied to prove that for mostsemilattice structures there exist a set $S=\{x_1, x_2, \ldots, x_n\}$ ofpositive integers and a real number $\alpha>0$ such that $S$ possesses thisstructure and the power LCM matrix $[[x_i, x_j]^\alpha]$ is singular.

Studying the singularity of LCM-type matrices via semilattice structures and their M\"obius functions