## Numerical SemigroupsLet N be the set of nonnegative integers. A numerical semigroup is a nonempty subset S of N that is closed under addition, contains the zero element, and whose complement in N is ?nite. If n ,...,n are positive integers with gcd{n ,...,n } = 1, then the set hn ,..., 1 e 1 e 1 n i = {? n +··· + ? n | ? ,...,? ? N} is a numerical semigroup. Every numer e 1 1 e e 1 e ical semigroup is of this form. The simplicity of this concept makes it possible to state problems that are easy to understand but whose resolution is far from being trivial. This fact attracted several mathematicians like Frobenius and Sylvester at the end of the 19th century. This is how for instance the Frobenius problem arose, concerned with ?nding a formula depending on n ,...,n for the largest integer not belonging to hn ,...,n i (see [52] 1 e 1 e for a nice state of the art on this problem). |

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### Contents

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5 | |

Numerical semigroups with maximal embedding dimension | 19 |

Irreducible numerical semigroups | 33 |

Proportionally modular numerical semigroups | 56 |

The quotient of a numerical semigroup by a positive integer | 77 |

Families of numerical semigroups closed under finite intersections and adjoin of the Frobenius number | 91 |

Presentations of a numerical semigroup | 105 |

The gluing of numerical semigroups | 123 |

Numerical semigroups with embedding dimension three | 137 |

The structure of a numerical semigroup | 155 |

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Apery set Archimedean Archimedean semigroup Arf numerical semigroup Assume ax mod Bezout's identity binary relation cardinality characterization compute congruence connected component consequence contradicting coprime Corollary deduce define denote Dickson's lemma embedding dimension three Example Exercise exists a positive Frobenius number Frobenius variety gluing graph greatest common divisor Hence hypothesis ical semigroup idempotents identity element implies integer greater intersection of finitely irreducible numerical semigroup isomorphic J. C. Rosales maximal embedding dimension minimal presentation minimal system modular numerical semigroup nonnegative integers numerical semi numerical semigroup minimally P. A. Garcıa-Sánchez positive integer Proof proportionally modular numerical Proposition 2.12 pseudo-symmetric numerical semigroup quasi-Archimedean R-classes rational numbers relatively prime result saturated numerical semigroup semigroup and let semigroup with embedding semigroup with m(S semigroup with multiplicity semigroups with maximal set of numerical Springer Science+Business Media subadditive function submonoid subset symmetric numerical semigroup Theorem torsion free view of Lemma