Gcd-closed sets and divisibility among power GCD matrices and power LCM matrices

Abstract

Let $a, b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. For any $x\in S$, let $G_{S}(x) :=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. In this paper, we first establish some interesting and important arithmetic properties of gcd-closed sets satisfying the condition $\mathcal{G}$ (i.e., for any element $x\in S$, either $G_S(x)$ contains at most one element, or $G_S(x)$ contains at least two elements and satisfies that $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$ for any $\{y_1,y_2\}\subseteq G_S(x)$), and then make use of these results to show that $(S^a)|(S^b)$ and $(S^a)|[S^b]$ when $a|b$ and $S$ is a gcd-closed set satisfying the condition $\mathcal{G}$. This confirms a conjecture of Hong proposed in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.} {\bf 113} (2026), 231-243].