Affine stratifications from finite mis\`ere quotients

TL;DR

This paper proves that fibers of morphisms from affine semigroups to commutative monoids have affine stratifications, supporting a conjecture about lattice games with finite misère quotients and advancing understanding of their combinatorial structure.

Contribution

It establishes that fibers of such morphisms always admit affine stratifications, using mesoprimary decomposition and new characterizations, thus confirming a special case of a conjecture related to lattice games.

Findings

Fibers possess affine stratifications as finite disjoint unions of translates of normal affine semigroups.

The proof employs mesoprimary decomposition of monoid congruences.

Supports a conjecture on affine stratifications for lattice games with finite misère quotients.

Abstract

Given a morphism from an affine semigroup Q to an arbitrary commutative monoid, it is shown that every fiber possesses an affine stratification: a partition into a finite disjoint union of translates of normal affine semigroups. The proof rests on mesoprimary decomposition of monoid congruences [arXiv:1107.4699] and a novel list of equivalent conditions characterizing the existence of an affine stratification. The motivating consequence of the main result is a special case of a conjecture due to Guo and the author [arXiv:0908.3473, arXiv:1105.5420] on the existence of affine stratifications for (the set of winning positions of) any lattice game. The special case proved here assumes that the lattice game has finite mis\'ere quotient, in the sense of Plambeck and Siegel [arXiv:math/0501315, arXiv:math/0609825v5].