Determinant and inverse of join matrices on two sets

Let (P,⪯) be a lattice and f a complex-valued function on P. We define meet and join matrices on two arbitrary subsets X and Y of P by (X,Y)f=(f(xi∧yj)) and [X,Y]f=(f(xi∨yj)) respectively. Here we present expressions for the determinant and the inverse of [X,Y]f. Our main goal is to cover the case when f is not semimultiplicative since the formulas presented earlier for [X,Y]f cannot be applied in this situation. In cases when f is semimultiplicative we obtain several new and known formulas for the determinant and inverse of (X,Y)f and the usual meet and join matrices (S)f and [S]f. We also apply these formulas to LCM, MAX, GCD and MIN matrices, which are special cases of join and meet matrices.