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An interesting class of subsets of lattice points in %-space arises naturally in the mathematical theory of (context free) languages. This is the class of semilinear subsets, a subset of lattice points being semilinear if it is the finite union of cosets of finitely generated sub-semigroups of the set of all lattice points with nonnegative coordinates.
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Ginsburg et al. (1966) studied this question.
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