GCD-closed sets and divisibility of Smith matrices
Let \(S\) be a set of \(n\) distinct positive integers. \textit{K. Bourque} and \textit{S. Ligh} [Linear Algebra Appl. 174, 65--74 (1992; Zbl 0761.15013)] showed that if \(S\) is factor closed, then the GCD matrix \((S)\) divides the LCM matrix \([S]\) in the ring of \(n\times n\) matrices over the integers. \textit{S. Hong} [Linear Algebra Appl. 345, No. 1--3, 225--233 (2002; Zbl 0995.15006)] showed that \((S)\mid [S]\) for any gcd closed set \(S\) with \(n\le 3\) and that for any \(n\ge 4\), there is a gcd closed set \(S\) with \(|S| = n\) such that \((S)\nmid [S]\). Hong also proposed the problem of characterizing all gcd closed sets \(S\) with \(|S|\ge 4\) such that \((S)\mid [S]\). For \(x\in S\), let \(G_S(x)\) denote the set of greatest-type divisors of \(x\) in \(S\). \textit{W. Feng} et al. [Discrete Math. 309, No. 9, 2627--2639 (2009; Zbl 1263.11042)] answered this problem when \(\max_{x\in S} |G_S (x)| \le 2\). \textit{J. Zhao} [Linear Multilinear Algebra 62, No. 6, 735--748 (2014; Zbl 1350.11042)] and \textit{E. Altınışık} et al. [Linear Algebra Appl. 516, 47--68 (2017; Zbl 1403.11022)] characterized all gcd closed sets \(S\) such that \((S)\mid [S]\) for \(4 \le |S| \le 7\) and \(|S| = 8\), respectively. The present authors introduce a new method to answer this problem in the case \(\max_{x\in S} |G_S (x)| = 3\). This problem remains open in the cases \(\max_{x\in S} |G_S (x)| \ge 4\) and \(|S| \ge 9\). \textit{I. Korkee} and \textit{P. Haukkanen} [Linear Algebra Appl. 429, No. 8--9, 1929--1943 (2008; Zbl 1157.11009)] considered a more general problem. They found all meet closed sets with at most five elements such that the associated meet matrix divides the associated join matrix
scientific article; zbMATH DE number 7481004
scientific article