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.