On Smithʹs determinant Original Research Article

We give a brief review of papers relating to Smithʹs determinant and point out a common structure that can be found in many extensions and analogues of Smithʹs determinant. We present the common structure in the language of posets. We also make an investigation on a conjecture of Beslin and Ligh on greatest common divisor (GCD) matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture. Further, we give a counterexample for the conjecture of Bourque and Ligh that the least common multiple matrix on any GCD-closed set is invertible.