Determinant and Inverse of Meet and Join Matrices

Abstract

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 .