On a general form of meet matrices associated with incidence functions

Let P = (P, <=, boolean AND) be a meet-semilattice with least element 0 such that every principal order ideal is finite. We define the function Psi on P x P by Psi(x, y) = (z <= x boolean AND y)Sigma f (0, z)g(z, x)h(z, y), where f, g and h are incidence functions of P. We calculate the determinant and the inverse of the matrix [Psi(x(i), x(j))], where S = {x(1), x(2),..., x(n)} is a meet-closed or lower-closed subset of P and h is an element of L-S. Here L-S is the class of incidence functions defined by L-S = {h\ (x(i), x(j) is an element of S, x <= x(i) <= x(j)) double right arrow h(x, x(i)) = h(x, x(j))}. We apply the results to meet matrices and obtain known determinant formulae and new inverse formulae for them. These results also concern GCD matrices, which are number-theoretic special cases of meet matrices. We also apply the results to the matrix [C(x(i), x(j))], where C(m, n) is the usual Ramanujan's sum.