Numerical Semigroups (eBook)

Contents 7
Introduction 10
Notable elements 13
Introduction 13
Monoids and monoid homomorphisms 13
Multiplicity and embedding dimension 14
Frobenius number and genus 17
Pseudo-Frobenius numbers 21
Exercises 24
Numerical semigroups with maximal embedding dimension 27
Introduction 27
Characterizations 28
Arf numerical semigroups 31
Saturated numerical semigroups 36
Exercises 39
Irreducible numerical semigroups 41
Introduction 41
Symmetric and pseudo-symmetric numerical semigroups 41
Irreducible numerical semigroups with arbitrary multiplicity and embedding dimension 46
Unitary extensions of a numerical semigroup 52
Decomposition of a numerical semigroup into irreducibles 55
Fundamental gaps of a numerical semigroup 59
Exercises 61
Proportionally modular numerical semigroups 64
Introduction 64
Periodic subadditive functions 64
The numerical semigroup associated to an intervalof rational numbers 66
Bézout sequences 68
Minimal generators of a proportionally modular numerical semigroup 72
Modular numerical semigroups 76
Opened modular numerical semigroups 78
Exercises 82
The quotient of a numerical semigroup by a positive integer 84
Introduction 84
Notable elements 85
One half of an irreducible numerical semigroup 86
Numerical semigroups having a Toms decomposition 90
Exercises 95
Families of numerical semigroups closed under finite intersections and adjoin of the Frobenius number 98
Introduction 98
The directed graph of the set of numerical semigroups 98
Frobenius varieties 100
Intersecting Frobenius varieties 105
Systems of generators with respect to a Frobenius variety 106
The directed graph of a Frobenius variety 107
Exercises 111
Presentations of a numerical semigroup 112
Introduction 112
Free monoids and presentations 113
Minimal presentations of a numerical semigroup 117
Computing minimal presentations 120
An upper bound for the cardinality of a minimal presentation 124
Exercises 127
The gluing of numerical semigroups 130
Introduction 130
The concept of gluing 131
Complete intersection numerical semigroups 134
Gluing of numerical semigroups 136
Free numerical semigroups 139
Exercises 142
Numerical semigroups with embedding dimension three 144
Introduction 144
Numerical semigroups with Apéry sets of unique expression 144
Irreducible numerical semigroups with embedding dimension three 148
Pseudo-Frobenius numbers and genus of an embedding dimension three numerical semigroup 154
Exercises 161
The structure of a numerical semigroup 162
Introduction 162
Levin's theorem 162
Structure theorem 166
N-monoids 169
Exercises 175
Bibliography 177
List of symbols 182
Index 184