# Generalized eigenvalue problems for meet and join matrices on semilattices

## TL;DR

This paper develops new bounds for generalized eigenvalues of meet and join matrices on semilattices, with applications to number-theoretic lattices, enhancing understanding of their spectral properties.

## Contribution

It introduces flexible methods to bound eigenvalues of meet and join matrices on semilattices, including sharper bounds for GCD and LCM matrices using divisor lattice properties.

## Key findings

- New bounds for eigenvalues of meet matrices
- Sharper bounds for GCD and LCM matrices
- Effective bounds demonstrated on number-theoretic lattices

## Abstract

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.