Divisibility among power GCD and power LCM matrices on certain gcd-closed sets
Jixiang Wan, Guangyan Zhu
TL;DR
The paper investigates divisibility relations among power GCD and power LCM matrices on gcd-closed sets, focusing on sets with $\max_{x\in S}|G_S(x)|=2$. It develops inverse and determinant formulas (via Bourque–Ligh and Hong) and introduces the condition $\mathcal{G}$ to control the greatest-type divisors, enabling a proof that if $a|b$ and $S$ is gcd-closed with $\mathcal{G}$, then $(S^a)|(S^b)$, $(S^a)|[S^b]$, and $[S^a]|[S^b]$ in $M_{|S|}({\mathbb Z})$. The work extends the Feng–Hong–Zhao theorem and partially confirms Hong’s conjecture, while also providing explicit gcd-closed examples where $\mathcal{G}$ fails yet the divisibility relations still hold for certain $b>1$, underscoring that $\mathcal{G}$ is not universally necessary. The results are established via a number-theoretic and combinatorial approach, leveraging determinant/inverse structure and case analyses, and they raise open questions about the necessity of $\mathcal{G}$ for all $b$.
Abstract
Let $(x, y)$ and $[x, y]$ denote the greatest common divisor and the least common multiple of the integers $x$ and $y$ respectively. We denote by $|T|$ the number of elements of a finite set $T$. Let $a,b$ and $n$ be positive integers and let $S=\{x_1, ..., x_n\}$ be a set of $n$ distinct positive integers. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix whose $(i,j)$-entry is the $a$th power of $(x_i,x_j)$ (resp. $[x_i,x_j]$). For any $x\in S$, define $G_{S}(x):=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. In this paper, we show that if $a|b$ and $S$ is gcd closed (namely, $(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i, j\le n$) and $\max_{x\in S}\{|G_S (x)|\}=2$ and the condition $\mathcal{G}$ being satisfied (i.e., any element $x\in S$ satisfies that either $|G_S(x)|\le 1$, or $G_S(x)=\{y_1,y_2\}$ satisfying that $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$), then $(S^a)|(S^b), (S^a)|[S^b]$ and $[S^a]|[S^b]$ hold in the ring $M_{n}({\bf Z})$. Furthermore, we show the existences of gcd-closed sets $S$ such that $S$ does not satisfy the condition $\mathcal{G}$ and such factorizations are true. Our result extends the Feng-Hong-Zhao theorem gotten in 2009. This also partially confirms a conjecture raised by Hong in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.}, doi:10.1017/S0004972725100361].