Determinant and Inverse of Meet and Join Matrices

We define meet and join matrices on two subsets X and Y of a lattice (P,≼) with respect to a complex-valued function f on P by (X,Y)f=(f(xi∧yi)) and [X,Y]f=(f(xi∨yi)), respectively. We present expressions for the determinant and the inverse of (X,Y)f and [X,Y]f, and as special cases we obtain several new and known formulas for the determinant and the inverse of the usual meet and join matrices (S)f and [S]f