Studying the singularity of LCM-type matrices via semilattice structures and their Möbius functions

The invertibility of Least Common Multiple (LCM) matrices and their Hadamard powers have been extensively studied over the years by many authors. Bourque and Ligh conjectured in 1992 that the LCM matrix [S]=[[xi,xj]][S]=[[xi,xj]] on any GCD closed set S={x1,x2,…,xn}S={x1,x2,…,xn} is invertible, but in 1997 this was proven to be false. Nevertheless, many open conjectures concerning LCM matrices and their Hadamard powers remain. In this paper we utilize lattice-theoretic structures and the Möbius function to explain the singularity of classical LCM matrices and their Hadamard powers. As a result we disprove some open conjectures of Hong. Elementary mathematical analysis is applied to prove that for most semilattice structures there exists a set S={x1,x2,…,xn}S={x1,x2,…,xn} of positive integers and a real number α>0α>0 such that S   possesses this structure and the power LCM matrix [[xi,xj]α][[xi,xj]α] is singular.