Equivariant lattice bases

We study lattices in free abelian groups of infinite rank that are invariant under the action of the infinite symmetric group, with emphasis on finiteness of their equivariant bases. Our framework provides a new method for proving finiteness results in algebraic statistics. As an illustration, we show that every invariant lattice in $\mathbb{Z}^{(\mathbb{N}\times[c])}$, where $c\in\mathbb{N}$, has a finite equivariant Graver basis. This result generalizes and strengthens several finiteness results about Markov bases in the literature.