Affine stratifications from finite mis`ere quotients

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].

💡 Research Summary

The paper investigates the structure of fibers of a morphism from an affine semigroup (Q) to an arbitrary commutative monoid (M). A fiber is the pre‑image (\varphi^{-1}(m)) of an element (m\in M) under a monoid homomorphism (\varphi:Q\to M). While such fibers can be highly intricate, the author proves that every fiber admits an affine stratification: it can be written as a finite disjoint union of translates (\tau_i+N_i) where each (N_i) is a normal affine semigroup (i.e., a saturated subsemigroup of a lattice) and each (\tau_i) is an element of the ambient group.

The core of the argument rests on mesoprimary decomposition of monoid congruences, a technique introduced in earlier work (arXiv:1107.4699). The congruence induced by (\varphi) on (Q) is decomposed into mesoprimary components, each of which corresponds to a normal affine semigroup together with a translation. By showing that the index of each mesoprimary component is finite, the author guarantees that the fiber splits into finitely many such pieces.

A novel contribution is a list of four equivalent conditions characterizing when a fiber possesses an affine stratification: (1) the fiber is finitely generated as a semigroup; (2) the difference of any two fibers is again an affine semigroup; (3) the intersection of a fiber with the ambient lattice has finite index; and (4) all mesoprimary components have finite index. Proving the equivalence of these conditions provides multiple practical criteria for detecting affine stratifications in concrete situations.

The motivation comes from lattice games, a class of impartial combinatorial games whose positions form a subset of (\mathbb{Z}^d). Guo and the author previously conjectured that the set of winning positions of any lattice game admits an affine stratification. This conjecture remains open in full generality, but the present paper resolves a significant special case: when the game has a finite misère quotient in the sense of Plambeck and Siegel (arXiv:math/0501315, arXiv:math/0609825v5). A finite misère quotient means that the misère equivalence classes of game positions form a finite monoid (M). The natural map (\varphi) sending each position to its equivalence class yields a fiber (\varphi^{-1}(0)) that exactly coincides with the set of (\mathscr{P})-positions (winning positions for the previous player). Applying the general affine‑stratification theorem to this fiber shows that the winning set is a finite disjoint union of translates of normal affine semigroups. Consequently, any lattice game with a finite misère quotient automatically satisfies the Guo–Miller conjecture.

Beyond the immediate game‑theoretic application, the results have broader implications for combinatorial commutative algebra and integer programming. An affine stratification provides a concrete, piecewise‑linear description of otherwise opaque semigroup fibers, enabling algorithmic enumeration, optimization, and geometric visualization. Moreover, the equivalence of the four criteria links algebraic properties (finite generation, mesoprimary finiteness) with geometric ones (finite index in a lattice), thereby enriching the toolkit for researchers working at the interface of monoid theory, polyhedral geometry, and combinatorial game theory.

The paper concludes with several directions for future work: extending the stratification result to morphisms whose induced congruence has infinite mesoprimary index; developing explicit algorithms based on mesoprimary decomposition for computing the strata; and exploring connections between affine stratifications and other algebraic invariants such as Hilbert bases, toric ideals, and modular forms. These avenues promise to deepen our understanding of how algebraic structures govern combinatorial dynamics in games and beyond.