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

TL;DR

This paper confirms a conjecture and completely solves an open problem regarding the divisibility of GCD matrices by LCM matrices on gcd-closed sets, extending previous work to cases where the set's structure is more complex.

Contribution

It proves the Zhao-Chen-Hong conjecture and provides a complete characterization of gcd-closed sets where GCD matrices divide LCM matrices, especially for cases with larger G_S(x) sets.

Findings

Confirmed the Zhao-Chen-Hong conjecture.

Provided a complete characterization for divisibility conditions.

Extended previous results to cases with larger G_S(x) sets.

Abstract

Let $S=\{x_1,\ldots, x_n\}$ 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 $\max_{x \in S}\{|G_S(x)|\} \leq 2$. 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.