Rosales / García-Sánchez Numerical Semigroups

1. Auflage 2009
ISBN: 978-1-4419-0160-6
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 20, 181 Seiten

Reihe: Developments in Mathematics

'Numerical Semigroups' is the first monograph devoted exclusively to the development of the theory of  numerical semigroups. This concise, self-contained text is accessible to first year graduate students, giving the full background needed for readers unfamiliar with the topic. Researchers will find the tools presented useful in producing examples and counterexamples in other fields such as algebraic geometry, number theory, and linear programming.

Autoren/Hrsg.

Rosales, J. C.

García-Sánchez, P. A.

Inhaltsverzeichnis

1;Contents;7
2;Introduction;10
3;Notable elements;13
3.1;Introduction;13
3.2;Monoids and monoid homomorphisms;13
3.3;Multiplicity and embedding dimension;14
3.4;Frobenius number and genus;17
3.5;Pseudo-Frobenius numbers;21
3.6;Exercises;24
4;Numerical semigroups with maximal embedding dimension;27
4.1;Introduction;27
4.2;Characterizations;28
4.3;Arf numerical semigroups;31
4.4;Saturated numerical semigroups;36
4.5;Exercises;39
5;Irreducible numerical semigroups;41
5.1;Introduction;41
5.2;Symmetric and pseudo-symmetric numerical semigroups;41
5.3;Irreducible numerical semigroups with arbitrary multiplicity and embedding dimension;46
5.4;Unitary extensions of a numerical semigroup;52
5.5;Decomposition of a numerical semigroup into irreducibles;55
5.6;Fundamental gaps of a numerical semigroup;59
5.7;Exercises;61
6;Proportionally modular numerical semigroups;64
6.1;Introduction;64
6.2;Periodic subadditive functions;64
6.3;The numerical semigroup associated to an intervalof rational numbers;66
6.4;Bézout sequences;68
6.5;Minimal generators of a proportionally modular numerical semigroup;72
6.6;Modular numerical semigroups;76
6.7;Opened modular numerical semigroups;78
6.8;Exercises;82
7;The quotient of a numerical semigroup by a positive integer;84
7.1;Introduction;84
7.2;Notable elements;85
7.3;One half of an irreducible numerical semigroup;86
7.4;Numerical semigroups having a Toms decomposition;90
7.5;Exercises;95
8;Families of numerical semigroups closed under finite intersections and adjoin of the Frobenius number;98
8.1;Introduction;98
8.2;The directed graph of the set of numerical semigroups;98
8.3;Frobenius varieties;100
8.4;Intersecting Frobenius varieties;105
8.5;Systems of generators with respect to a Frobenius variety;106
8.6;The directed graph of a Frobenius variety;107
8.7;Exercises;111
9;Presentations of a numerical semigroup;112
9.1;Introduction;112
9.2;Free monoids and presentations;113
9.3;Minimal presentations of a numerical semigroup;117
9.4;Computing minimal presentations;120
9.5;An upper bound for the cardinality of a minimal presentation;124
9.6;Exercises;127
10;The gluing of numerical semigroups;130
10.1;Introduction;130
10.2;The concept of gluing;131
10.3;Complete intersection numerical semigroups;134
10.4;Gluing of numerical semigroups;136
10.5;Free numerical semigroups;139
10.6;Exercises;142
11;Numerical semigroups with embedding dimension three;144
11.1;Introduction;144
11.2;Numerical semigroups with Apéry sets of unique expression;144
11.3;Irreducible numerical semigroups with embedding dimension three;148
11.4;Pseudo-Frobenius numbers and genus of an embedding dimension three numerical semigroup;154
11.5;Exercises;161
12;The structure of a numerical semigroup;162
12.1;Introduction;162
12.2;Levin's theorem;162
12.3;Structure theorem;166
12.4;N-monoids;169
12.5;Exercises;175
13;Bibliography;177
14;List of symbols;182
15;Index;184