ON SMITH DETERMINANT

We give a brief review of papers relating to Smith's determinant and p oint out a common structure that can be found in many extensions and a nalogues of Smith's determinant. We present the common structure in th e language of posets. We also make an investigation on a conjecture of Beslin and Ligh on greatest common divisor (GCD) matrices in the sens e 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-c losed set is invertible. (C) Elsevier Science Inc., 1997.