Studying the divisibility of power LCM matrics 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 $|S|\ge 4$ such that the GCD matrix $(S)$ divides the LCM matrix $[S]$ in the ring $M_n(\mathbb{Z})$. For $x\in S,$ let $G_S(x):=\{z\in S:z<x,z|x\text{ and }(z|y|x,y\in S)\Rightarrow y\in \{z,x\}\}$. In 2009, Feng, Hong and Zhao answered this problem in the context where ${\max }_{x\in S}\{|G_S(x)|\}\le 2$. In 2022, Zhao, Chen and Hong obtained a necessary and sufficient condition on the gcd-closed set $S$ with ${\max }_{x\in S}\{|G_S(x)|\}=3$ such that $(S)|\left[S\right].$ Meanwhile, they raised a conjecture on the necessary and sufficient condition such that $(S)|\left[S\right]$ holds for the remaining case ${\max }_{x\in S}\{|G_S(x)|\}\ge 4$. In this papar, we confirm the Zhao-Chen-Hong conjecture from a novel perspective, consequently solve Hong's open problem completely.