Abstract
Given a morphism from an affine semigroup 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 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 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 misere quotient, in the sense of Plambeck and Siegel.
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