On a general form of join matrices associated with incidence functions

Summary.

Let $(P,\leq) = (P, \vee)$ be a join-semilattice with finite principal order filters and let $\Psi_{\vee}$ denote the function on P  × P defined by

$$\Psi_{\vee}(x, y) = \sum\limits_{x\vee y\leq z} f(x, z)g(y,z)h(z),$$

where f and g are incidence functions of P and h is a complex-valued function on P. We calculate the determinant and the inverse of the matrix $[\Psi_{\vee}(x_i, x_j)]$, where S = {x ₁, x ₂,...,x _n } is a join-closed subset of P and (x _i , x _j ∈S, x _i  ≤ x _j  ≤ z) ⇒ f(x _i , z) = f(x _j , z) holds for all z ∈ P.

As special cases we obtain formulae for join matrices $([S]_h)_{ij} = h(x_i \vee x_j)$ . The determinant formulae obtained for join matrices are known in the literature, whereas the inverse formulae are new. We also obtain new results for LCM and LCUM matrices, which are number-theoretic special cases of join matrices.