Studying the divisibility of power LCM matrices by power GCD matrices on gcd-closed sets

Abstract

Let $S=\{x_1,\dots ,x_n\}$ be a gcd-closed set (i.e. $(x_i,x_j)\in S$ for all $1\le i,j\le n$). In 2002, Hong proposed the divisibility problem of characterizing all gcd-closed sets S with $\left|S\right|\ge 4$ such that the GCD matrix (S) divides the LCM matrix $\left[S\right]$ in the ring $M_n\left(Z\right)$. For $x\in S$, let $G_S\left(x\right):=\{z\in S:z<x,z|x\text{ and }\left(z\right|y|x,y\in S)\Rightarrow y\in \{z,x\}\}$. In 2009, Feng, Hong and Zhao answered this problem in the context where ${\max }_{x\in S}\left\{\right|G_S\left(x\right)\left|\right\}\le 2$. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set S with ${\max }_{x\in S}\left\{\right|G_S\left(x\right)\left|\right\}=3$ such that $\left(S\right)|\left[S\right]$. Meanwhile, they raised a conjecture on the necessary and sufficient condition such that $\left(S\right)|\left[S\right]$ holds for the remaining case ${\max }_{x\in S}\left\{\right|G_S\left(x\right)\left|\right\}\ge 4$. In this paper, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.