Determinants of matrices associated with incidence functions on posets

Let S = {x(1),...,x(n)} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let [f (x(i) Lambda x(j))] denote the n x n matrix having f evaluated at the meet x(i) Lambda x(j) of x(i) and x(j) as its i, j-entry and [f (x(i) V x(j))] denote the n x n matrix having f evaluated at the join x(i) V x(j) of x(i) and x(j) as its i, j-entry. The set S is said to be meet-closed if x(i) Lambda x(j) is an element of S for all 1 less than or equal to i, j less than or equal to n. In this paper we get explicit combinatorial formulas for the determinants of matrices [f (x(i) A x(j))] and [f (x(i) V x(j))] on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices f (x(i) A x(j))] and [f (x(i) V x(j))] on any meet-closed set S to be nonsingular. Finally, we give some numbertheoretic applications.