Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals This paper is an investigation of chains of lattice ideals in polynomial rings that are invariant under a symmetric group action. The polynomial rings are increasing in Krull dimension. Underlying many computations is the fact that in Noetherian rings (such as ℂ[x 1 ,⋯,x n ]) any ascending chain of ideals I 1 ⊆I 2 ⊆⋯ eventually stabilizes. However, the chains in question will not stabilize in the Noetherian sense. Using properties of nice orderings, the authors show that invariant chains of Laurent lattice ideals stabilize up to monomial localization. Moreover, for specific Laurent toric ideals, the authors give an algorithm for constructing the stabilization generators. The algorithm has been implemented using the Macaulay2 package FourTiTwo. The family of toric ideals studied have applications to algebraic statistics. The authors do a great job of providing relevant references within the area and to the applications in algebraic statistics. Throughout the paper, the authors effectively use a running example to demonstrate their approaches and algorithm. The paper is closed with a number of interesting open problems.