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). |
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 |
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a₁ Ap S,n Apéry set Archimedean Archimedean semigroup Arf numerical semigroup Assume ax mod b₁ Bézout sequence binary relation cardinality characterization complete intersection compute congruence connected component Corollary deduce define denote embedding dimension three Exercise exists a positive Free(X Frobenius number Frobenius variety gcd{a gluing graph greatest common divisor Hence ical semigroup idempotents identity element implies integer greater irreducible numerical semigroup isomorphic J. C. Rosales Math maximal embedding dimension minimal presentation minimal system modular numerical semigroup n₁ nonnegative integers numerical semi numerical semigroup minimally P. A. García-Sánchez PF(S positive integer Proof proportionally modular numerical Proposition 2.12 pseudo-symmetric numerical semigroup quasi-Archimedean R-classes rational numbers relatively prime saturated numerical semigroup semigroup and let Semigroup Forum semigroup with embedding semigroup with m(S semigroup with multiplicity semigroups with maximal set of numerical SG(S Springer Science+Business Media subadditive function submonoid subset symmetric numerical semigroup Theorem torsion free