Divisibility among power GCD matrices and among power LCM matrices on finitely many coprime divisor chains

Let a,b and h be positive integers and S={x1,...,xh} be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation σ on {1,...,h} such that xσ(1)|...|xσ(h). We say that the set S consists of finitely many coprime divisor chains if there is a positive integer k such that we can partition S as S=S1∪...∪Sk, where all the Si are divisor chains and each element of Si is coprime to each element of Sj for 1⩽i≠j⩽k. The matrix having the ath power (xi,xj)a of the greatest common divisor of xi and xj as its (i,j)-entry is called ath power greatest common divisor (GCD) matrix defined on S, denoted by (Sa). Similarly we can define the ath power LCM matrix [Sa]. In this paper, we show that if a|b and S consists of finitely many coprime divisor chains with 1∈S, then in the ring Mh(Z) of h×h matrices over integers, we have (Sa)|(Sb),[Sa]|[Sb] and (Sa)|[Sb]. But such results fail to be true if a|b. These results confirm partially Hong’s conjectures raised in 2008.