Numerical Semigroups and Applications [book]

Abdallah Assi, Pedro A. García-Sánchez
2016 RSME Springer Series  
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
more » ... dulo m, and we conclude that we can choose A to have finitely many elements.
doi:10.1007/978-3-319-41330-3 fatcat:yhoounfrjrcidc74rqw4tkwiie