  This paper lies in the intersection of several fields: number theory, lattice
theory, multilinear algebra, and scientific computing. We adapt existing
solution algorithms for tensor eigenvalue problems to the tensor-train
framework. As an application, we consider eigenvalue problems associated with a
class of lattice-theoretic meet and join tensors, which may be regarded as
multidimensional extensions of the classically studied meet and join matrices
such as GCD and LCM matrices, respectively. In order to effectively apply the
solution algorithms, we show that meet tensors have an explicit low-rank
tensor-train decomposition with sparse tensor-train cores with respect to the
dimension. Moreover, this representation is independent of tensor order, which
eliminates the so-called curse of dimensionality from the numerical analysis of
these objects and makes the solution of tensor eigenvalue problems tractable
with increasing dimensionality and order. For LCM tensors it is shown that a
tensor-train decomposition with an a priori known TT rank exists under certain
assumptions. We present a series of easily reproducible numerical examples
covering tensor eigenvalue and generalized eigenvalue problems that serve as
future benchmarks. The numerical results are used to assess the sharpness of
existing theoretical estimates.
