  We study chains of lattice ideals that are invariant under a symmetric group
action. In our setting, the ambient rings for these ideals are polynomial rings
which are increasing in (Krull) dimension. Thus, these chains will fail to
stabilize in the traditional commutative algebra sense. However, we prove a
theorem which says that "up to the action of the group", these chains locally
stabilize. We also give an algorithm, which we have implemented in software,
for explicitly constructing these stabilization generators for a family of
Laurent toric ideals involved in applications to algebraic statistics. We close
with several open problems and conjectures arising from our theoretical and
computational investigations.
