Numerical Semigroups

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Springer Science & Business Media, Dec 24, 2009 - Mathematics - 181 pages
Let 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

Introduction
2
Notable elements
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
Bibliography
171
List of symbols
177
Index
179
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