Periodic weakly-cancellative semigroups.
Let be an abelian semigroup, and a finite subset of . The sumset consists of all sums of elements of , with repetitions allowed. Let denote the cardinality of . Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial such that for all sufficiently large . Lattice point counting is also used to prove that sumsets of the form have multivariate polynomial growth.