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

Abstract

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.