Gcd-closed sets and divisibility of Smith matrices

Abstract

Let n be a positive integer and S={x1,...,xn} a set of n distinct positive integers. In 1992, Bourque and Ligh showed that if S is factor closed (i.e. d∈S if d|x for any x∈S), then the GCD matrix (S) divides the LCM matrix [S] in the ring of n×n matrices over the integers. In 2002, Hong showed that such factorization holds for any gcd-closed set S (i.e. (xi,xj)∈S for all 1≤i,j≤n) with |S|≤3. But for any integer n≥4, there is a gcd-closed set S with |S|=n such that (S)∤[S]. Meanwhile, Hong proposed the problem of characterizing all gcd-closed sets S with |S|≥4 such that (S)|[S]. For x∈S, let GS(x):={d∈S|d<x,d|xand(d|y|x,y∈S)⇒y∈{d,x}}. In 2009, Feng, Hong and Zhao answered this problem when maxx∈S{|GS(x)|}≤2. Zhao in 2014 and Altinisik, Yildiz and Keskin in 2017 characterized all gcd-closed sets S such that (S)|[S] for the case 4≤|S|≤7 and |S|=8, respectively. In this paper, we introduce a new method to study Hong's problem. We obtain a necessary and sufficient condition on the gcd-closed set S with maxx∈S{|GS(x)|}=3 such that (S)|[S]. This solves partially Hong's problem.