Affine stratification conjecture for squarefree lattice games

A lattice game is a combinatorial game played on lattice positions, and a game is squarefree when its relevant move structure satisfies the squarefree condition. An affine stratification is a finite disjoint union decomposition of the set of positions into translates of affine semigroups.

Affine stratification conjecture. Every squarefree lattice game possesses an affine stratification.

Such a stratification is intended to provide a finite algebraic description of the game's position sets and thereby support effective algorithms for determining winning positions and moves. The paper also states the claim in the broader contexts of normal and misère play; the corresponding unrestricted claim for all lattice games was disproved.