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Numerical Semigroups and Applications
[book]

2016
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RSME Springer Series
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Proposition 3. Every numerical semigroup is finitely generated. Proof. Let A be a system of generators of S (S itself is a system of generators). Let m be the multiplicity of S. Clearly m ∈ A. Assume that a < a ′ are two elements in A such that a ≡ a ′ mod m. Then a ′ = km + a for some positive integer k. So we can remove a ′ from A and we still have a generating system for S. Observe that at the end of this process we have at most one element in A in each congruence class modulo m, and we

doi:10.1007/978-3-319-41330-3
fatcat:yhoounfrjrcidc74rqw4tkwiie