On Smith's determinant (Q1359193)

The authors give a brief review of papers relating to Smith's determinant [cf. \textit{H. J. S. Smith}, Proc. Lond. Math. Soc. 7, 208--212 (1876; JFM 08.0074.02)] and point out a common structure that can be found in many extensions and analogues of Smith's determinant. The common structure is presented in the language of posets. The authors also make an investigation on a conjecture of \textit{S. Beslin} and \textit{S. Ligh} [Bull. Aust. Math. Soc. 40, No. 3, 413--415 (1989; Zbl 0675.10002); Fibonacci Q. 30, No. 2, 157--160 (1992; Zbl 0752.11012)] on greatest common divisor (GCD) matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture. Further, a counterexample for the conjecture of \textit{K. Bourque} and \textit{S. Ligh} [Linear Algebra Appl. 174, 65--74 (1992; Zbl 0761.15013)] that the least common multiple matrix on any GCD-closed set is invertible is given.