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. $\mathrm{l}\mathrm{c}\mathrm{m}(x_i,x_j{)}^a$) as its $(i,j)$-entry. For any integer $x\in S$, if $(y<x,y|z|x\mathrm{a}\mathrm{n}\mathrm{d}y,z\in S)\Rightarrow z\in \{y,x\}$, 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 $\underset{x\in S}{max}\{|\{y\in S:y\text{is a greatest-type divisor of}x\mathrm{i}\mathrm{n}S\}|\}=2$.