New results on the divisibility of power GCD and power LCM matrices

Guangyan Zhu; Mao Li; Xiaofan Xu; Guangyan Zhu; Mao Li; Xiaofan Xu

doi:10.3934/math.20221003

AIMS Mathematics

2022, Volume 7, Issue 10: 18239-18252. doi: 10.3934/math.20221003

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Research article

New results on the divisibility of power GCD and power LCM matrices

1.
School of Teacher Education, Hubei Minzu University, Enshi 445000, China
2.
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
3.
School of Big Data and Statistics, Sichuan Tourism University, Chengdu 610100, China

Received: 30 April 2022 Revised: 01 July 2022 Accepted: 08 July 2022 Published: 11 August 2022
MSC : 11A05, 11C20, 15B36

Let $ a, b $ and $ n $ be positive integers and let $ S $ be a set consisting of $ n $ distinct positive integers $ x_1, ..., x_{n-1} $ and $ x_n $. Let $ (S^a) $ (resp. $ [S^a] $) denote the $ n\times n $ matrix having $ \gcd(x_i, x_j)^a $ (resp. $ {\rm lcm}(x_i, x_j)^a $) as its $ (i, j) $-entry. For any integer $ x\in S $, if $ (y < x, y|z|x \ {\rm and} \ y, z\in S)\Rightarrow z\in\{y, x\} $, then $ y $ is called a greatest-type divisor of $ x $ in $ S $. In this paper, we establish some results about the divisibility between $ (S^a) $ and $ (S^b) $, between $ (S^a) $ and $ [S^b] $ and between $ [S^a] $ and $ [S^b] $ when $ a|b $, $ S $ is gcd closed (i.e., $ \gcd(x_i, x_j)\in S $ for all $ 1\le i, j\le n $), and $ \max_{x\in S}\{|\{y\in S: y \ \text{is a greatest-type divisor of} \ x \ {\rm in} \ S\}|\} = 2 $.
- divisibility,
- power matrix,
- greatest-type divisor,
- gcd-closed set
Citation: Guangyan Zhu, Mao Li, Xiaofan Xu. New results on the divisibility of power GCD and power LCM matrices[J]. AIMS Mathematics, 2022, 7(10): 18239-18252. doi: 10.3934/math.20221003

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Abstract

Let $ a, b $ and $ n $ be positive integers and let $ S $ be a set consisting of $ n $ distinct positive integers $ x_1, ..., x_{n-1} $ and $ x_n $. Let $ (S^a) $ (resp. $ [S^a] $) denote the $ n\times n $ matrix having $ \gcd(x_i, x_j)^a $ (resp. $ {\rm lcm}(x_i, x_j)^a $) as its $ (i, j) $-entry. For any integer $ x\in S $, if $ (y < x, y|z|x \ {\rm and} \ y, z\in S)\Rightarrow z\in\{y, x\} $, then $ y $ is called a greatest-type divisor of $ x $ in $ S $. In this paper, we establish some results about the divisibility between $ (S^a) $ and $ (S^b) $, between $ (S^a) $ and $ [S^b] $ and between $ [S^a] $ and $ [S^b] $ when $ a|b $, $ S $ is gcd closed (i.e., $ \gcd(x_i, x_j)\in S $ for all $ 1\le i, j\le n $), and $ \max_{x\in S}\{|\{y\in S: y \ \text{is a greatest-type divisor of} \ x \ {\rm in} \ S\}|\} = 2 $.

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