On the divisibility of meet and join matrices Original Research Article

Let (P,less-than-or-equals, slant)=(P,logical and,logical or) be a lattice, let S={x1,x2,…,xn} be a meet-closed subset of P and let image be a function. We characterize the matrix divisibility of the join matrix [S]f=[f(xilogical orxj)] by the meet matrix (S)f=[f(xilogical andxj)] in the ring image in terms of the usual divisibility in image, and we present two algorithms for constructing certain classes of meet-closed sets S such that (S)f divides [S]f. As an example we present the lattice-theoretic structure of all meet-closed sets with at most five elements possessing the matrix divisibility property. Finally, we show that our methods solve some open problems in the divisor lattice, concerning the divisibility of GCD and LCM matrices.