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ON THE DIVISIBILITY AMONG POWER LCM MATRICES ON GCD-CLOSED SETS

Published online by Cambridge University Press:  19 May 2022

GUANGYAN ZHU*
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, PR China and School of Teacher Education, Hubei Minzu University, Enshi 445000, PR China
MAO LI
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, PR China e-mail: [email protected]
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Abstract

Let $a,b$ and n be positive integers and let $S=\{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. For ${x\in S}$ , define $G_{S}(x)=\{d\in S: d<x, \,d\mid x \ \mathrm {and} \ (d\mid y\mid x, y\in S)\Rightarrow y\in \{d,x\}\}$ . Denote by $[S^a]$ the $n\times n$ matrix having the ath power of the least common multiple of $x_i$ and $x_j$ as its $(i,j)$ -entry. We show that the bth power matrix $[S^b]$ is divisible by the ath power matrix $[S^a]$ if $a\mid b$ and S is gcd closed (that is, $\gcd (x_i, x_j)\in S$ for all integers i and j with $1\le i, j\le n$ ) and $\max _{x\in S} \{|G_S (x)|\}=1$ . This confirms a conjecture of Shaofang Hong [‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 1001–1008].

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

1 Introduction

For arbitrary integers x and y, we denote by $(x,y)$ the greatest common divisor of x and y and by $[x,y]$ their least common multiple. Let $a, b$ and n be positive integers. Let $S = \{x_1, \ldots , x_n\}$ be a set of n distinct positive integers. Let $\xi _a$ be the arithmetic function defined by $\xi _a=x^a$ for any positive integer x. Let $(S^a)$ and $[S^a]$ stand for the $n\times n$ matrices whose $(i,j)$ -entry is $\xi _a((x_i, x_j))$ and $\xi _a([x_i, x_j])$ respectively. We call $(S^a)$ the $ath$ power GCD matrix and $[S^a]$ the $ath$ power LCM matrix. The set S is factor closed (FC) if $(x\in S, d\mid x)\Rightarrow d\in S$ and gcd closed if $(x_i, x_j)\in S$ for all integers i and j with $1\le i,j\le n$ . Obviously, an FC set must be gcd closed but the converse is not true. Nearly 150 years ago, Smith [Reference Smith15] proved that

(1.1) $$ \begin{align} \det([x_i,x_j])=\prod_{k=1}^n\varphi(x_k)\pi(x_k) \end{align} $$

if S is FC, where $\varphi $ is Euler’s totient function and $\pi $ is the multiplicative function defined for the prime power $p^r$ by $\pi (p^r)=-p$ . There are many generalisations of Smith’s determinant (1.1) and related results (see, for instance, [Reference Altinisik, Yildiz and Keskin1Reference Korkee and Haukkanen14, Reference Tan and Li16Reference Zhu, Cheng and Zhao21]). In particular, an elegant result was achieved by Hong et al. [Reference Hong, Hu and Lin8] stating that for any integer $n\ge 2$ ,

$$ \begin{align*}\det([i,j])_{2\le i,j\le n}=\bigg(\prod_{k=1}^n \varphi (k)\pi(k)\bigg)\sum_{\substack{t=1\\ t \text{ is square free}}}^n\frac{t\mu(t)}{\varphi (t)}, \end{align*} $$

where $\mu $ is the M $\ddot {\textrm {o}}$ bius function and an integer $x\ge 1$ is called square free if x is not divisible by the square of any prime number.

As usual, $\mathbb {Z}$ and $|S|$ denote the ring of integers and the cardinality of the set S. Hong [Reference Hong9] introduced the concept of greatest-type divisor when he solved the Bourque–Ligh conjecture. For any integer $x\in S$ , y is called a greatest-type divisor of x if

$$ \begin{align*}(y<x, \,y\mid z\mid x \ \mathrm{and} \ y,z\in S)\Rightarrow z\in\{y,x\}.\end{align*} $$

Let $ G_S(x):=\{y\in S: y \ \text {is a greatest-type divisor of} \ x \ \mathrm {in} \ S\} $ and let $M_n(\mathbb {Z})$ stand for the ring of $n\times n$ matrices over the integers. Bourque and Ligh [Reference Bourque and Ligh4] proved that $(S)$ divides $[S]$ in the ring $M_n(\mathbb {Z})$ (that is, $[S]=B(S)$ or $[S]=(S)B$ for some $B\in M_n(\mathbb {Z})$ ) if S is FC. Hong [Reference Hong10] showed that such a factorisation is not true when S is gcd closed and $\max _{x \in S}\{|{G_S(x)}|\}=2$ . The results of Bourque–Ligh and Hong were generalised by Korkee and Haukkanen [Reference Korkee and Haukkanen14] and by Chen et al. [Reference Chen, Feng, Hong and Qiu6]. Feng et al. [Reference Feng, Hong and Zhao7], Zhao [Reference Zhao17], Altinisik et al. [Reference Altinisik, Yildiz and Keskin1] and Zhao et al. [Reference Zhao, Chen and Hong18] used the concept of greatest-type divisor to characterise the gcd-closed sets S with $\max _{x \in S}\{|{G_S(x)}|\}\le 3$ such that $(S^a)\mid [S^a]$ which partially solved an open problem of Hong [Reference Hong10].

Hong [Reference Hong12] investigated divisibility among power GCD matrices and among power LCM matrices. It was proved in [Reference Hong12] that $(S^a)\mid (S^b), (S^a) \mid [S^b]$ and $[S^a] \mid [S^b]$ if $a \mid b$ and S is a divisor chain (that is, $x_{\sigma (1)}| \cdots |x_{\sigma (n)}$ for a permutation $\sigma $ of $\{1,\ldots , n\}$ ), and such factorisations are no longer true if $a\nmid b$ and $|S|\ge 2$ . Evidently, a divisor chain is gcd closed but not conversely. Recently, Zhu [Reference Zhu19] confirmed two conjectures of Hong raised in [Reference Hong12] stating that if $a\mid b$ and S is a gcd-closed set with $\max _{x\in S}\{|G_S(x)|\}=1$ , then both the bth power GCD matrix $(S^b)$ and the bth power LCM matrix $[S^b]$ are divisible by the ath power GCD matrix $(S^a)$ . At the end of [Reference Hong12], Hong also conjectured that if $a \mid b$ and $S=\{x_1,\ldots , x_n \}$ is gcd closed and $\max _{x\in S}\{|G_S(x)|\}=1$ , then $[S^a] \mid [S^b]$ in the ring $M_n(\mathbb {Z})$ . Tan and Li [Reference Tan and Li16] partially confirmed this conjecture by proving that $[S^a] \mid [S^b]$ in the ring $M_{|S|}(\mathbb {Z})$ if $a \mid b$ and S consists of finitely many coprime divisor chains with $1\in S$ and that such a divisibility relation is not true if $a\nmid b$ . However, the conjecture still remains open.

Our goal is to present a proof of Hong’s conjecture. The main result of the paper is the following theorem.

Theorem 1.1. If a and b are positive integers such that $a \mid b$ and S is a gcd-closed set such that $\max _{x\in S}\{|G_S(x)|\}=1$ , then the ath power LCM matrix $[S^a]$ divides the bth power LCM matrix $[S^b]$ in the ring $M_{|S|}(\mathbb {Z})$ .

The proof of Theorem 1.1 is similar to that of Feng et al. [Reference Feng, Hong and Zhao7] in character, but it is more complicated. This paper is organised as follows. In Section 2, we supply several preliminary lemmas needed in the proof of Theorem 1.1. Section 3 is devoted to the proof of Theorem 1.1.

One can easily check that for any permutation $\sigma $ on the set $\{1, \ldots , n \}$ , $[S^a]\mid [S^b]\Leftrightarrow [S_{\sigma }^a]\mid [S_{\sigma } ^b]$ , where $S_\sigma :=\{x_{\sigma (1)},\ldots , x_{\sigma (n)}\}$ . Without loss of any generality, we can always assume that the set $S=\{x_1,\ldots , x_n \}$ satisfies $x_1<\cdots <x_n$ .

2 Auxiliary results

In this section, we provide several lemmas that will be needed in the proof of Theorem 1.1. We begin with a result due to Hong which gives the formula for the determinant of the power LCM matrix on a gcd-closed set.

Lemma 2.1 [Reference Hong11, Lemma 2.1].

If S is gcd closed, then

(2.1) $$ \begin{align} \det[S^a]=\prod\limits_{k=1}^n x_k^{2a}\alpha_{a,k}, \end{align} $$

where

(2.2) $$ \begin{align} \alpha_{a,k}:=\sum_{\substack{d|x_k\\ d\nmid x_t,\,x_t<x_k}} \bigg(\dfrac{1}{\xi_a}*\mu\bigg)(d) \end{align} $$

and ${1}/{\xi _a}$ is the arithmetic function defined for any positive integer x by ${({1}/{\xi _a})(x):= x^{-a}}$ .

Lemma 2.2 [Reference Bourque and Ligh5, Theorem 3].

If S is a gcd-closed set and $(f((x_i,x_j)))$ is invertible, then $ (f((x_i,x_j)))^{-1}=(a_{ij}), $ where

$$ \begin{align*}a_{ij}:=\sum_{\substack{x_i|x_k\\ x_j|x_k}} \dfrac{c_{ik}c_{jk}}{\delta_k}\end{align*} $$

with

(2.3) $$ \begin{align} \delta_k:= \sum_{\substack{d|x_k\\ d\nmid x_t,\, x_t<x_k}} (f*\mu)(d) \quad\mbox{and}\quad c_{ij}:=\sum _{\substack{dx_i|x_j\\ dx_i\nmid x_t,\, x_t<x_j}} \mu(d). \end{align} $$

Lemma 2.3 [Reference Hong11, Lemma 2.3].

Let m be a positive integer. Then

$$ \begin{align*}{\underset{d|m}{\sum}}\bigg(\dfrac{1}{\xi_a}*\mu \bigg)(d)=m^{-a}. \end{align*} $$

Lemma 2.4 [Reference Feng, Hong and Zhao7, Lemma 2.2].

Let S be gcd closed and $\max _{x\in S}\{|G_S(x)|\}=1$ . Let $\alpha _{a,k}$ be defined as in (2.2). If $G_S(x_k)=\{x_{k_1}\}$ for $2\le k\le |S|$ , then $\alpha _{a,k}=x_k^{-a}-x_{k_1}^{-a}.$

Lemma 2.5. Let S be gcd closed and $\max _{x\in S}\{|G_S(x)|\}=1$ . Let $\alpha _{a,k}$ and $c_{ij}$ be defined as in (2.2) and (2.3), respectively. Then $[S^a]$ is nonsingular and $[S^a]^{-1}=(s_{ij})_{1\le i,j\le n}$ with

$$ \begin{align*} s_{ij}:=\dfrac{1}{x_i^a x_j^a} \sum_{\substack{x_i|x_k\\ x_j|x_k}} \dfrac{c_{ik}c_{jk}}{\alpha_{a,k}}. \end{align*} $$

Proof. Since ${[x_i,x_j]}^a={x_i^ax_j^a}/{{(x_i,x_j)}^a}$ ,

(2.4) $$ \begin{align} [S^a]=D\bigg(\dfrac{1}{\xi_a}(x_i,x_j)\bigg)D, \end{align} $$

where $D:=\mathrm {diag}(x_1^a,\ldots ,x_n^a)$ . By (2.1) and (2.4),

$$ \begin{align*} \det\bigg(\dfrac{1}{\xi_a}((x_i,x_j))\bigg) =\prod\limits_{k=1}^n\alpha_{a,k}. \end{align*} $$

By Lemma 2.3, $\alpha _{a,1}=x_1^{-a}$ . For $2\le k\le n$ , since $\max _{x\in S}\{|G_S(x)|\}=1$ , one may let $G_S(x_k)=\{x_{k_1}\}$ . By Lemma 2.4, $\alpha _{a,k}=x_k^{-a}-x_{k_1}^{-a}\neq 0$ . So the matrix $(({1/\xi _a})((x_i,x_j)))$ is nonsingular. Now applying Lemma 2.2 gives

(2.5) $$ \begin{align} \bigg(\dfrac{1}{\xi_a}((x_i,x_j))\bigg)^{-1}=(h_{ij}), \end{align} $$

where

$$ \begin{align*}h_{ij}:=\sum_{\substack{x_i|x_k\\ x_j|x_k}} \dfrac{c_{ik}c_{jk}}{\alpha_{a,k}}.\end{align*} $$

The desired result follows immediately from (2.4) and (2.5).

We next recall some basic results on gcd-closed sets.

Lemma 2.6 [Reference Feng, Hong and Zhao7, Lemma 2.3].

Let S be a gcd-closed set with $|S|\ge 2$ . Let $c_{ij}$ be defined as in (2.3). Then

$$ \begin{align*}c_{w1}=\left\{\begin{aligned} {1}& \quad \mbox{if } w=1,\\ {0}&\quad \mbox{otherwise}. \end{aligned} \right. \end{align*} $$

Further, if $G_S(x_m)=\{x_{m_1}\}$ for $2\le m\le |S|$ , then

$$ \begin{align*}c_{wm}=\left\{ \begin{aligned} {-1}&\quad \mbox{if }w=m_1,\\ {1}&\quad \mbox{if } w=m,\\ {0}&\quad \mbox{otherwise}. \end{aligned} \right. \end{align*} $$

Lemma 2.7 [Reference Feng, Hong and Zhao7, Lemma 3.1].

Let S be gcd closed and $x,z\in S$ such that $x\nmid z$ . If ${G_S(x)=\{y\}}$ , then $(x, z)=(y, z)$ .

Lemma 2.8. Let S be gcd closed and $x, y\in S$ with $G_S(x)=\{y\}$ . If $a\mid b$ , then for any $z, r\in S$ with $r\mid x$ , $y^a[z,x]^b-x^a[z,y]^b$ is divisible by each of $x^a(y^a-x^a)$ and $r^a(y^a-x^a)$ .

Proof. We divide the proof into two cases.

Case 1: $x\nmid z$ . By Lemma 2.7, $(x, z)=(y, z)$ , which implies

(2.6) $$ \begin{align} y^a[z,x]^b-x^a[z,y]^b=y^a\dfrac{z^bx^b} {(z,x)^b}-x^a\dfrac{z^by^b}{(z,y)^b} =\dfrac{z^b}{(z,x)^b}x^ay^a(x^{b-a}-y^{b-a}). \end{align} $$

Since $a\mid b$ ,

$$ \begin{align*}x^{b-a}-y^{b-a}=(x^a-y^a)\sum _{i=0}^{{({b}/{a})}-2} (x^a)^{{({b}/{a})}-2-i}y^{ai} \quad\mbox{and}\quad \sum _{i=0}^{{({b}/{a})}-2} (x^a)^{{({b}/{a})}-2-i}y^{ai}\in \mathbb{Z}. \end{align*} $$

Hence, $(x^a-y^a)\mid (x^{b-a}-y^{b-a})$ . Then by (2.6), we deduce that $y^a[z,x]^b-x^a[z,y]^b$ is divisible by each of $x^a(y^a-x^a)$ and $r^a(y^a-x^a)$ .

Case 2: $x\mid z$ . Then $[x, z]=[y, z]=z$ . It follows that

$$ \begin{align*}y^a[z,x]^b-x^a[z,y]^b=y^az^b-x^az^b=z^b(y^a-x^a).\end{align*} $$

Since $a\mid b$ , the desired results follow immediately.

Lemma 2.9. Let S be gcd closed and $\max _{x\in S}\{|G_S(x)|\}=1$ . If $a\mid b$ , then all the elements of the nth column and the nth row of $[S^b][S^a]^{-1}$ are integers.

Proof. The proof of Lemma 2.9 is divided into two cases.

Case 1: $1\le i\le n$ and $j=n$ . By Lemmas 2.5 and 2.6,

$$ \begin{align*} {([S^b][S^a]^{-1})}_{in} =&\sum_{m=1}^n[x_i, x_m]^b\dfrac{1}{x_m^ax_n^a}\sum_{\substack{x_m|x_k\\ x_n|x_k}} \dfrac{c_{mk}c_{nk}}{\alpha _{a,k}}\\ =&\dfrac{1}{x_n^a}\sum_{m=1}^n\dfrac{[x_i,x_m]^bc_{mn}}{x_m^a\alpha_{a,n}} =\dfrac{1}{x_n^a\alpha_{a,n}}\sum_{m=1}^n\dfrac{[x_i,x_m]^bc_{mn}}{x_m^a}. \end{align*} $$

Since $\max _{x\in S}\{|G_S(x)|\}=1$ , we may let $G_S(x_n)=\{x_{n_1}\}$ . Then by Lemmas 2.4, 2.6 and 2.8,

$$ \begin{align*}{([S^b])[S^a]^{-1})}_{in} =\dfrac{x_{n_1}^a[x_i,x_n]^b-x_n^a[x_i,x_{n_1}]^b}{x_n^a(x_{n_1}^a-x_n^a)}\in \mathbb{Z} \end{align*} $$

as required.

Case 2: $i=n,\ 1\le j\le n-1$ . Then

$$ \begin{align*} {( [S^b][S^a]^{-1})}_{nj} =\sum_{m=1}^n[x_n, x_m]^b\dfrac{1}{x_m^ax_j^a}\sum_{\substack{x_m|x_k\\ x_j|x_k}} \frac {c_{mk}c_{jk}}{\alpha _{a,k}} =\sum_{x_j|x_k}\frac{c_{jk}}{x_j^a\alpha _{a,k}}\sum_{x_m|x_k}\dfrac{1}{x_m^a}c_{mk}[x_m,x_n]^b. \end{align*} $$

We claim that

$$ \begin{align*}\gamma_k:={\frac{1}{x_j^a\alpha _{a,k}}\sum_{x_m|x_k}\dfrac{1}{x_m^a}c_{mk}[x_m,x_n]^b}\in \mathbb{Z}\end{align*} $$

for any positive integer k with $x_j\mid x_k$ .

If $k=1$ , then $m=j=1$ . In this case,

$$ \begin{align*}\gamma_1=\dfrac{1}{\alpha _{a,1}}\cdot\dfrac{1} {x_1^{2a}}\cdot c_{11}\cdot[x_1,x_n]^b =\dfrac{[x_1,x_n]^b}{x_1^a} =\dfrac{x_1^{b-a}x_n^b}{{(x_1,x_n)}^b}\in \mathbb{Z}.\end{align*} $$

Now let $k>1$ . We can set $G_S(x_k)=\{x_{k_1}\}$ since $|G_S(x_k)|=1$ . By Lemmas 2.4, 2.6 and 2.8,

$$ \begin{align*}\gamma_k={\frac{1}{x_j^a\alpha _{a,k}}\sum_{x_m|x_k} \dfrac{1}{x_m^a}c_{mk}[x_m,x_n]^b} ={\frac{x^a_{k_1}[x_k,x_n]^b-x_k^a[x_{k_1},x_n]^b} {x_j^a(x_{k_1}^a-x_k^a)}}\in \mathbb{Z} \end{align*} $$

as desired. This concludes the proof of the claim and of Lemma 2.9.

Finally, we can use Lemma 2.9 to establish the main result of this section.

Lemma 2.10. Let S be gcd closed and $\max _{x\in S}\{|G_S(x)|\}=1$ . Let $S_1:=S\setminus \{x_n\}=\{x_1, \ldots , x_{n-1}\}$ . If $a\mid b$ , then $[S^b][S^a]^{-1}\in M_n(\mathbb {Z})$ if and only if $[S_1^b][S_1^a]^{-1}\in M_{n-1}(\mathbb {Z}).$

Proof. First, it follows from the hypothesis and Lemma 2.9 that all the elements of the nth column and the nth row of $[S^b][S^a]^{-1}$ are integers. So it suffices to show that

(2.7) $$ \begin{align} \mathcal{A}_{ij}:={([S^b][S^a]^{-1})}_{ij} -{([S_1^b][S_1^a]^{-1})}_{ij}\in \mathbb{Z} \end{align} $$

for all integers i and j with $1\le i,j\le n-1$ .

To see this, define

$$ \begin{align*} e_{uv}:=\left\{ \begin{aligned} 1&\quad \mbox{if} \ x_v\mid x_u,\\ 0&\quad \mbox{if} \ x_v\nmid x_u, \end{aligned} \right. \end{align*} $$

for all integers u and v between 1 and n. Then $e_{nj}=1$ if $x_j\mid x_n$ and $e_{nj}=0$ otherwise. Furthermore, for any integer m with $1\le m\le n-1$ , one has $e_{nm}=1$ if $x_m\mid x_n$ and $e_{nm}=0$ otherwise. We then deduce that

(2.8) $$ \begin{align} \mathcal{A}_{ij} & =\sum_{m=1}^n[x_i,x_m]^b\sum_{\substack{x_m|x_k\\ x_j|x_k}} \frac{c_{mk}c_{jk}}{x_m^ax_j^a\alpha _{a,k}} -\sum_{m=1}^{n-1}[x_i,x_m]^b\sum _{\substack{x_m|x_k\\ x_j|x_k,\, x_k\neq x_n}} \frac{c_{mk}c_{jk}}{x_m^ax_j^a\alpha _{a,k}}\notag\\ & =\frac{c_{nn}c_{jn}}{x_n^ax_j^a\alpha _{a,n}}[x_i,x_n]^b e_{nj}+ \sum_{m=1}^{n-1}\frac{c_{mn}c_{jn}}{x_m^ax_j^a\alpha _{a,n}} [x_i,x_m]^b e_{nj}e_{nm} \notag\\ & =e_{nj}\frac{c_{jn}}{x_j^a\alpha _{a,n}} \bigg(\dfrac{[x_i,x_n]^b}{x_n^a}+\sum_{m=1}^{n-1} \dfrac{[x_i,x_m]^bc_{mn}e_{nm}}{x_m^a} \bigg) :=e_{nj} A_{ij}. \end{align} $$

Let us now show that $A_{ij}\in \mathbb {Z}$ . Since $\max _{x\in S}\{|G_S(x)|\}=1$ , one may let $G_S(x_n)=\{x_{n_1}\}$ . From Lemma 2.4, $\alpha _{a,n}=x_n^{-a}-x_{n_1}^{-a}.$ However, by Lemma 2.6, for any integer m with $1\le m\le n-1$ , $c_{mn}=-1$ if $m=n_1$ and $c_{mn}=0$ otherwise. It follows from (2.8) and Lemma 2.8 that

(2.9) $$ \begin{align} {A_{ij}=\frac{x_{n_1}^a[x_i,x_n]^b-x_n^a[x_i,x_{n_1}]^b} {x_j^a(x_{n_1}^a-x_n^a)}\cdot c_{jn}}\in \mathbb{Z}. \end{align} $$

Since $e_{nj}\in \{0,1\}$ , (2.8) and (2.9) yield (2.7).

The proof of Lemma 2.10 is complete.

3 Proof of Theorem 1.1

We prove Theorem 1.1 by using induction on $n=|S|$ .

For $n=1$ , the statement is clearly true.

Let $n=2$ . Since $S=\{x_1, x_2\}$ is gcd closed, $(x_1, x_2)=x_1$ and $x_1\mid x_2$ . It follows that

$$ \begin{align*}\begin{aligned} {[S^b]}[S^a]^{-1}&= \begin{pmatrix} x_1^b&x_2^b\\[3pt] x_2^b&x_2^b \end{pmatrix} \cdot\frac {1}{x_2^a(x_1^a-x_2^a)} \begin{pmatrix} x_2^a&-x_2^a\\[3pt] -x_2^a&x_1^a \end{pmatrix} =\begin{pmatrix} \mathcal{B}&-x_1^a\mathcal{C}\\[3pt] 0&x_2^{b-a} \end{pmatrix}, \end{aligned} \end{align*} $$

where

$$ \begin{align*}\mathcal{B}:=\dfrac {x_2^b-x_1^b}{x_2^a-x_1^a} \quad\mathrm{and}\quad \mathcal{C}:=\dfrac {x_2^{b-a}-x_1^{b-a}}{x_2^a-x_1^a}.\end{align*} $$

Since $a\mid b$ , implying that $a\mid (b-a)$ , it follows that $\mathcal {B}\in \mathbb {Z}$ and $\mathcal {C}\in \mathbb {Z}$ , that is, ${[S^b]}[S^a]^{-1}\in M_2(\mathbb {Z})$ . The statement is true for this case.

Let $n=3$ . Since $S=\{x_1,x_2,x_3\}$ is gcd closed, we have $x_1\mid x_i\ (i=2, 3)$ and $(x_2,x_3)=x_1$ or $x_2$ . Consider the following two cases.

Case 1: $(x_2,x_3)=x_1$ . Then one computes

$$ \begin{align*} \begin{split} {[S^b]}[S^a]^{-1} &=\begin{pmatrix} x_1^b&x_2^b&x_3^b\\[4pt] x_2^b&x_2^b&\dfrac{x_2^bx_3^b}{x_1^b}\\[4pt] x_3^b&\dfrac{x_2^bx_3^b}{x_1^b}&x_3^b \end{pmatrix} \cdot \frac {x_1^a}{x_2^ax_3^a(x_2^a-x_1^a)(x_3^a-x_1^a)}\\[4pt] &\quad \times \begin{pmatrix} \dfrac{x_1^{2a}x_2^ax_3^a-x_2^{2a}x_3^{2a}}{x_1^{2a}} &\dfrac{x_2^ax_3^{2a}-x_1^ax_2^ax_3^a}{x_1^a} &\dfrac{x_2^{2a}x_3^a-x_1^ax_2^ax_3^a}{x_1^a}\\[4pt] \dfrac{x_2^ax_3^{2a}-x_1^ax_2^ax_3^a}{x_1^a}&x_1^ax_3^a-x_3^{2a}&0\\[4pt] \dfrac{x_2^{2a}x_3^a-x_1^ax_2^ax_3^a}{x_1^a}&0&x_1^ax_2^a-x_2^{2a} \end{pmatrix}\\ &=\begin{pmatrix} \mathcal{B}+x_3^a\mathcal{F}&-x_1^a\mathcal{C}&-x_1^a\mathcal{F}\\[4pt] x_3^a\mathcal{D}\mathcal{F}&x_2^{b-a}&-x_1^a\mathcal{D}\mathcal{F}\\[4pt] x_2^a\mathcal{E}\mathcal{C}&-x_1^a\mathcal{E}\mathcal{C}&x_3^{b-a} \end{pmatrix}, \end{split} \end{align*} $$

where $\mathcal {B}$ and $\mathcal {C}$ are as given earlier in this section, $\mathcal {D}:={x_2^b}/{x_1^b}$ , $\mathcal {E}:={x_3^b}/{x_1^b}$ and $\mathcal {F}:=({x_3^{b-a}-x_1^{b-a})}/({x_3^a-x_1^a}).$ Since $x_1\mid x_2$ , $x_1\mid x_3$ and $a\mid (b-a)$ , all of $\mathcal {B}, \mathcal {C}, \mathcal {D}, \mathcal {E}$ and $\mathcal {F}$ are integers. Hence, ${[S^b]}[S^a]^{-1}\in M_3(\mathbb {Z})$ . The statement holds in this case.

Case 2: $(x_2,x_3)=x_2$ . Then $x_2\mid x_3$ . We compute

$$ \begin{align*}\begin{aligned} {[S^b]}[S^a]^{-1}&= \begin{pmatrix} x_1^b&x_2^b&x_3^b\\[4pt] x_2^b&x_2^b&x_3^b\\[4pt] x_3^b&x_3^b&x_3^b \end{pmatrix} \cdot \frac {1}{x_3^a(x_2^a-x_1^a)(x_3^a-x_2^a)}\\ &\quad \times \begin{pmatrix} x_3^a(x_2^a-x_3^a)&x_3^a(x_3^a-x_2^a)&0\\[4pt] x_3^a(x_3^a-x_2^a)&x_3^a(x_1^a-x_3^a)&x_3^a(x_2^a-x_1^a)\\[4pt] 0&x_3^a(x_2^a-x_1^a)&x_2^a(x_1^a-x_2^a) \end{pmatrix}\\ &=\begin{pmatrix} \mathcal{B}&-\mathcal{B}+\mathcal{G}&-x_2^a\mathcal{H}\\[4pt] 0&\mathcal{G}&-x_2^a\mathcal{H}\\[4pt] 0&0&x_3^{b-a} \end{pmatrix}, \end{aligned} \end{align*} $$

where $\mathcal {B}$ is as before, $\mathcal {G}:={({x_3^b-x_2^b)}/{(x_3^a-x_2^a})}$ and $\mathcal {H}:={(x_3^{b-a}-x_2^{b-a})}/{(x_3^a-x_2^a)}$ . Since $a\mid b$ and $a\mid (b-a)$ imply that $\mathcal {G}\in \mathbb {Z}$ and $\mathcal {H}\in \mathbb {Z}$ , it follows immediately that ${[S^b]}[S^a]^{-1}\in M_3(\mathbb {Z})$ . The statement is true for this case.

Now let $n\ge 4$ . Assume that the statement is true for the $n-1$ case. In what follows, we show that the statement is true for the n case. Since S is gcd closed and $\max _{x\in S}\{|G_S(x)|\}=1$ , it follows that $S_1:=\{x_1,\ldots , x_{n-1}\}$ is also gcd closed and $\max _{x\in S_1}\{|G_{S_1}(x)|\}=1$ . Hence by the inductive hypothesis, $[S_1^b][S_1^a]^{-1}\in M_{n-1}( \mathbb {Z})$ . Finally, from Lemma 2.10, $[S^b][S^a]^{-1}\in M_n(\mathbb {Z})$ as desired.

This finishes the proof of Theorem 1.1. $\Box $

Acknowledgement

The authors would like to thank the anonymous referee for careful reading of the paper and helpful suggestions that improved its presentation.

References

Altinisik, E., Yildiz, M. and Keskin, A., ‘Non-divisibility of LCM matrices by GCD matrices on gcd-closed sets’, Linear Algebra Appl. 516 (2017), 4768.CrossRefGoogle Scholar
Bege, A., ‘Generalized LCM matrices’, Publ. Math. Debrecen 79 (2011), 309315.CrossRefGoogle Scholar
Beslin, S. and Ligh, S., ‘Another generalization of Smith’s determinant’, Bull. Aust. Math. Soc. 40 (1989), 413415.CrossRefGoogle Scholar
Bourque, K. and Ligh, S., ‘On GCD and LCM matrices’, Linear Algebra Appl. 174 (1992), 6574.CrossRefGoogle Scholar
Bourque, K. and Ligh, S., ‘Matrices associated with arithmetical functions’, Linear Multilinear Algebra 34 (1993), 261267.CrossRefGoogle Scholar
Chen, L., Feng, Y. L., Hong, S. F. and Qiu, M., ‘On the divisibility of matrices associated with multiplicative functions’, Publ. Math. Debrecen 100 (2022), 323335.CrossRefGoogle Scholar
Feng, W. D., Hong, S. F. and Zhao, J. R., ‘Divisibility properties of power LCM matrices by power GCD matrices on gcd-closed sets’, Discrete Math. 309 (2009), 26272639.CrossRefGoogle Scholar
Hong, S. A., Hu, S. N. and Lin, Z. B., ‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372382.CrossRefGoogle Scholar
Hong, S. F., ‘On the Bourque–Ligh conjecture of least common multiple matrices’, J. Algebra 218 (1999), 216228.CrossRefGoogle Scholar
Hong, S. F., ‘On the factorization of LCM matrices on gcd-closed sets’, Linear Algebra Appl. 345 (2002), 225233.CrossRefGoogle Scholar
Hong, S. F., ‘Notes on power LCM matrices’, Acta Arith. 111 (2004), 165177.CrossRefGoogle Scholar
Hong, S. F., ‘Divisibility properties of power GCD matrices and power LCM matrices’, Linear Algebra Appl. 428 (2008), 10011008.CrossRefGoogle Scholar
Hong, S. F., Zhao, J. R. and Yin, Y. Z., ‘Divisibility properties of Smith matrices’, Acta Arith. 132 (2008), 161175.CrossRefGoogle Scholar
Korkee, I. and Haukkanen, P., ‘On the divisibility of meet and join matrices’, Linear Algebra Appl. 429 (2008), 19291943.CrossRefGoogle Scholar
Smith, H. J. S., ‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. (3) 7 (1875), 208212.CrossRefGoogle Scholar
Tan, Q. R. and Li, M., ‘Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains’, Linear Algebra Appl. 438 (2013), 14541466.CrossRefGoogle Scholar
Zhao, J. R., ‘Divisibility of power LCM matrices by power GCD matrices on gcd-closed sets’, Linear Multilinear Algebra 62 (2014), 735748.10.1080/03081087.2013.786717CrossRefGoogle Scholar
Zhao, J. R., Chen, L. and Hong, S. F., ‘Gcd-closed sets and divisibility of Smith matrices’, J. Combin. Theory Ser. A 188 (2022), Article no. 105581, 23 pages.CrossRefGoogle Scholar
Zhu, G. Y., ‘On the divisibility among power GCD and power LCM matrices on gcd-closed sets’, Int. J. Number Theory, to appear.Google Scholar
Zhu, G. Y., ‘On a certain determinant for a U.F.D.’, Colloq. Math., to appear.Google Scholar
Zhu, G. Y., Cheng, K. M. and Zhao, W., ‘Notes on Hong’s conjecture on nonsingularity of power LCM matrices’, AIMS Math. 7 (2022), 1027610285.CrossRefGoogle Scholar