  We study generalized eigenvalue problems for meet and join matrices with
respect to incidence functions on semilattices. We provide new bounds for
generalized eigenvalues of meet matrices with respect to join matrices under
very general assumptions. The applied methodology is flexible, and it is shown
in the case of GCD and LCM matrices that even sharper bounds can be obtained by
applying the known properties of the divisor lattice. These results can also be
easily modified for the dual problem of eigenvalues of join matrices with
respect to meet matrices, which we briefly consider as well. We investigate the
effectiveness of the obtained bounds for select examples involving
number-theoretical lattices.
