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

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 $.