\contentsline {chapter}{\numberline {1}\leavevmode {\color {Chapter } Introduction }}{6}{chapter.1}
\contentsline {chapter}{\numberline {2}\leavevmode {\color {Chapter } Numerical Semigroups }}{9}{chapter.2}
\contentsline {section}{\numberline {2.1}\leavevmode {\color {Chapter } Generating Numerical Semigroups }}{9}{section.2.1}
\contentsline {subsection}{\numberline {2.1.1}\leavevmode {\color {Chapter }NumericalSemigroup}}{9}{subsection.2.1.1}
\contentsline {subsection}{\numberline {2.1.2}\leavevmode {\color {Chapter }ModularNumericalSemigroup}}{10}{subsection.2.1.2}
\contentsline {subsection}{\numberline {2.1.3}\leavevmode {\color {Chapter }ProportionallyModularNumericalSemigroup}}{10}{subsection.2.1.3}
\contentsline {subsection}{\numberline {2.1.4}\leavevmode {\color {Chapter }NumericalSemigroupByGenerators}}{11}{subsection.2.1.4}
\contentsline {section}{\numberline {2.2}\leavevmode {\color {Chapter }Some basic tests}}{11}{section.2.2}
\contentsline {subsection}{\numberline {2.2.1}\leavevmode {\color {Chapter }IsNumericalSemigroup}}{11}{subsection.2.2.1}
\contentsline {subsection}{\numberline {2.2.2}\leavevmode {\color {Chapter }RepresentsSmallElementsOfNumericalSemigroup}}{12}{subsection.2.2.2}
\contentsline {subsection}{\numberline {2.2.3}\leavevmode {\color {Chapter }RepresentsGapsOfNumericalSemigroup}}{12}{subsection.2.2.3}
\contentsline {subsection}{\numberline {2.2.4}\leavevmode {\color {Chapter }IsAperyListOfNumericalSemigroup}}{13}{subsection.2.2.4}
\contentsline {subsection}{\numberline {2.2.5}\leavevmode {\color {Chapter }IsSubsemigroupOfNumericalSemigroup}}{13}{subsection.2.2.5}
\contentsline {subsection}{\numberline {2.2.6}\leavevmode {\color {Chapter }BelongsToNumericalSemigroup}}{13}{subsection.2.2.6}
\contentsline {chapter}{\numberline {3}\leavevmode {\color {Chapter } Basic operations with numerical semigroups }}{15}{chapter.3}
\contentsline {section}{\numberline {3.1}\leavevmode {\color {Chapter } The definitions }}{15}{section.3.1}
\contentsline {subsection}{\numberline {3.1.1}\leavevmode {\color {Chapter }MultiplicityOfNumericalSemigroup}}{15}{subsection.3.1.1}
\contentsline {subsection}{\numberline {3.1.2}\leavevmode {\color {Chapter }GeneratorsOfNumericalSemigroup}}{15}{subsection.3.1.2}
\contentsline {subsection}{\numberline {3.1.3}\leavevmode {\color {Chapter }SmallElementsOfNumericalSemigroup}}{16}{subsection.3.1.3}
\contentsline {subsection}{\numberline {3.1.4}\leavevmode {\color {Chapter }FirstElementsOfNumericalSemigroup}}{16}{subsection.3.1.4}
\contentsline {subsection}{\numberline {3.1.5}\leavevmode {\color {Chapter }AperyListOfNumericalSemigroupWRTElement}}{16}{subsection.3.1.5}
\contentsline {subsection}{\numberline {3.1.6}\leavevmode {\color {Chapter }DrawAperyListOfNumericalSemigroup}}{16}{subsection.3.1.6}
\contentsline {subsection}{\numberline {3.1.7}\leavevmode {\color {Chapter }AperyListOfNumericalSemigroupAsGraph}}{17}{subsection.3.1.7}
\contentsline {section}{\numberline {3.2}\leavevmode {\color {Chapter }Frobenius Number}}{17}{section.3.2}
\contentsline {subsection}{\numberline {3.2.1}\leavevmode {\color {Chapter }FrobeniusNumberOfNumericalSemigroup}}{17}{subsection.3.2.1}
\contentsline {subsection}{\numberline {3.2.2}\leavevmode {\color {Chapter }FrobeniusNumber}}{17}{subsection.3.2.2}
\contentsline {subsection}{\numberline {3.2.3}\leavevmode {\color {Chapter }PseudoFrobeniusOfNumericalSemigroup}}{17}{subsection.3.2.3}
\contentsline {section}{\numberline {3.3}\leavevmode {\color {Chapter }Gaps}}{18}{section.3.3}
\contentsline {subsection}{\numberline {3.3.1}\leavevmode {\color {Chapter }GapsOfNumericalSemigroup}}{18}{subsection.3.3.1}
\contentsline {subsection}{\numberline {3.3.2}\leavevmode {\color {Chapter }FundamentalGapsOfNumericalSemigroup}}{18}{subsection.3.3.2}
\contentsline {subsection}{\numberline {3.3.3}\leavevmode {\color {Chapter }SpecialGapsOfNumericalSemigroup}}{18}{subsection.3.3.3}
\contentsline {chapter}{\numberline {4}\leavevmode {\color {Chapter } Presentations of Numerical Semigroups }}{19}{chapter.4}
\contentsline {section}{\numberline {4.1}\leavevmode {\color {Chapter }Presentations of Numerical Semigroups}}{19}{section.4.1}
\contentsline {subsection}{\numberline {4.1.1}\leavevmode {\color {Chapter }FortenTruncatedNCForNumericalSemigroups}}{19}{subsection.4.1.1}
\contentsline {subsection}{\numberline {4.1.2}\leavevmode {\color {Chapter }MinimalPresentationOfNumericalSemigroup}}{19}{subsection.4.1.2}
\contentsline {subsection}{\numberline {4.1.3}\leavevmode {\color {Chapter }GraphAssociatedToElementInNumericalSemigroup}}{20}{subsection.4.1.3}
\contentsline {chapter}{\numberline {5}\leavevmode {\color {Chapter } Constructing numerical semigroups from others }}{21}{chapter.5}
\contentsline {section}{\numberline {5.1}\leavevmode {\color {Chapter } Adding and removing elements of a numerical semigroup }}{21}{section.5.1}
\contentsline {subsection}{\numberline {5.1.1}\leavevmode {\color {Chapter }RemoveMinimalGeneratorFromNumericalSemigroup}}{21}{subsection.5.1.1}
\contentsline {subsection}{\numberline {5.1.2}\leavevmode {\color {Chapter }AddSpecialGapOfNumericalSemigroup}}{21}{subsection.5.1.2}
\contentsline {subsection}{\numberline {5.1.3}\leavevmode {\color {Chapter }IntersectionOfNumericalSemigroups}}{22}{subsection.5.1.3}
\contentsline {subsection}{\numberline {5.1.4}\leavevmode {\color {Chapter }QuotientOfNumericalSemigroup}}{22}{subsection.5.1.4}
\contentsline {section}{\numberline {5.2}\leavevmode {\color {Chapter } Constructing the set of all numerical semigroups containing a given numerical semigroup }}{23}{section.5.2}
\contentsline {subsection}{\numberline {5.2.1}\leavevmode {\color {Chapter }OverSemigroupsNumericalSemigroup}}{23}{subsection.5.2.1}
\contentsline {subsection}{\numberline {5.2.2}\leavevmode {\color {Chapter }NumericalSemigroupsWithFrobeniusNumber}}{23}{subsection.5.2.2}
\contentsline {chapter}{\numberline {6}\leavevmode {\color {Chapter } Irreducible numerical semigroups }}{24}{chapter.6}
\contentsline {section}{\numberline {6.1}\leavevmode {\color {Chapter } Irreducible numerical semigroups }}{24}{section.6.1}
\contentsline {subsection}{\numberline {6.1.1}\leavevmode {\color {Chapter }IsIrreducibleNumericalSemigroup}}{24}{subsection.6.1.1}
\contentsline {subsection}{\numberline {6.1.2}\leavevmode {\color {Chapter }IsSymmetricNumericalSemigroup}}{25}{subsection.6.1.2}
\contentsline {subsection}{\numberline {6.1.3}\leavevmode {\color {Chapter }IsPseudoSymmetricNumericalSemigroup}}{25}{subsection.6.1.3}
\contentsline {subsection}{\numberline {6.1.4}\leavevmode {\color {Chapter }AnIrreducibleNumericalSemigroupWithFrobeniusNumber}}{25}{subsection.6.1.4}
\contentsline {subsection}{\numberline {6.1.5}\leavevmode {\color {Chapter }IrreducibleNumericalSemigroupsWithFrobeniusNumber}}{25}{subsection.6.1.5}
\contentsline {subsection}{\numberline {6.1.6}\leavevmode {\color {Chapter }DecomposeIntoIrreducibles}}{26}{subsection.6.1.6}
\contentsline {chapter}{\numberline {7}\leavevmode {\color {Chapter } Ideals of numerical semigroups }}{27}{chapter.7}
\contentsline {section}{\numberline {7.1}\leavevmode {\color {Chapter } Ideals of numerical semigroups }}{27}{section.7.1}
\contentsline {subsection}{\numberline {7.1.1}\leavevmode {\color {Chapter }IdealOfNumericalSemigroup}}{27}{subsection.7.1.1}
\contentsline {subsection}{\numberline {7.1.2}\leavevmode {\color {Chapter }IsIdealOfNumericalSemigroup}}{27}{subsection.7.1.2}
\contentsline {subsection}{\numberline {7.1.3}\leavevmode {\color {Chapter }MinimalGeneratingSystemOfIdealOfNumericalSemigroup}}{28}{subsection.7.1.3}
\contentsline {subsection}{\numberline {7.1.4}\leavevmode {\color {Chapter }GeneratorsOfIdealOfNumericalSemigroup}}{28}{subsection.7.1.4}
\contentsline {subsection}{\numberline {7.1.5}\leavevmode {\color {Chapter }AmbientNumericalSemigroupOfIdeal}}{28}{subsection.7.1.5}
\contentsline {subsection}{\numberline {7.1.6}\leavevmode {\color {Chapter }SmallElementsOfIdealOfNumericalSemigroup}}{29}{subsection.7.1.6}
\contentsline {subsection}{\numberline {7.1.7}\leavevmode {\color {Chapter }BelongsToIdealOfNumericalSemigroup}}{29}{subsection.7.1.7}
\contentsline {subsection}{\numberline {7.1.8}\leavevmode {\color {Chapter }SumIdealsOfNumericalSemigroup}}{30}{subsection.7.1.8}
\contentsline {subsection}{\numberline {7.1.9}\leavevmode {\color {Chapter }MultipleOfIdealOfNumericalSemigroup}}{30}{subsection.7.1.9}
\contentsline {subsection}{\numberline {7.1.10}\leavevmode {\color {Chapter }SubtractIdealsOfNumericalSemigroup}}{30}{subsection.7.1.10}
\contentsline {subsection}{\numberline {7.1.11}\leavevmode {\color {Chapter }DifferenceOfIdealsOfNumericalSemigroup}}{31}{subsection.7.1.11}
\contentsline {subsection}{\numberline {7.1.12}\leavevmode {\color {Chapter }TranslationOfIdealOfNumericalSemigroup}}{31}{subsection.7.1.12}
\contentsline {subsection}{\numberline {7.1.13}\leavevmode {\color {Chapter }HilbertFunctionOfIdealOfNumericalSemigroup}}{32}{subsection.7.1.13}
\contentsline {subsection}{\numberline {7.1.14}\leavevmode {\color {Chapter }BlowUpIdealOfNumericalSemigroup}}{32}{subsection.7.1.14}
\contentsline {subsection}{\numberline {7.1.15}\leavevmode {\color {Chapter }ReductionNumberIdealNumericalSemigroup}}{32}{subsection.7.1.15}
\contentsline {subsection}{\numberline {7.1.16}\leavevmode {\color {Chapter }MaximalIdealOfNumericalSemigroup}}{33}{subsection.7.1.16}
\contentsline {subsection}{\numberline {7.1.17}\leavevmode {\color {Chapter }BlowUpOfNumericalSemigroup}}{33}{subsection.7.1.17}
\contentsline {subsection}{\numberline {7.1.18}\leavevmode {\color {Chapter }MicroInvariantsOfNumericalSemigroup}}{33}{subsection.7.1.18}
\contentsline {subsection}{\numberline {7.1.19}\leavevmode {\color {Chapter }IsGradedAssociatedRingNumericalSemigroupCM}}{34}{subsection.7.1.19}
\contentsline {subsection}{\numberline {7.1.20}\leavevmode {\color {Chapter }CanonicalIdealOfNumericalSemigroup}}{34}{subsection.7.1.20}
\contentsline {subsection}{\numberline {7.1.21}\leavevmode {\color {Chapter }IntersectionIdealsOfNumericalSemigroup}}{35}{subsection.7.1.21}
\contentsline {chapter}{\numberline {8}\leavevmode {\color {Chapter } Numerical semigroups with maximal embedding dimension }}{36}{chapter.8}
\contentsline {section}{\numberline {8.1}\leavevmode {\color {Chapter } Numerical semigroups with maximal embedding dimension }}{36}{section.8.1}
\contentsline {subsection}{\numberline {8.1.1}\leavevmode {\color {Chapter }IsMEDNumericalSemigroup}}{36}{subsection.8.1.1}
\contentsline {subsection}{\numberline {8.1.2}\leavevmode {\color {Chapter }MEDNumericalSemigroupClosure}}{37}{subsection.8.1.2}
\contentsline {subsection}{\numberline {8.1.3}\leavevmode {\color {Chapter }MinimalMEDGeneratingSystemOfMEDNumericalSemigroup}}{37}{subsection.8.1.3}
\contentsline {section}{\numberline {8.2}\leavevmode {\color {Chapter } Numerical semigroups with the Arf property and Arf closures }}{37}{section.8.2}
\contentsline {subsection}{\numberline {8.2.1}\leavevmode {\color {Chapter }IsArfNumericalSemigroup}}{37}{subsection.8.2.1}
\contentsline {subsection}{\numberline {8.2.2}\leavevmode {\color {Chapter }ArfNumericalSemigroupClosure}}{38}{subsection.8.2.2}
\contentsline {subsection}{\numberline {8.2.3}\leavevmode {\color {Chapter }MinimalArfGeneratingSystemOfArfNumericalSemigroup}}{38}{subsection.8.2.3}
\contentsline {chapter}{\numberline {9}\leavevmode {\color {Chapter } Catenary and Tame degrees of numerical semigroups }}{39}{chapter.9}
\contentsline {section}{\numberline {9.1}\leavevmode {\color {Chapter } Factorizations in Numerical Semigroups }}{39}{section.9.1}
\contentsline {subsection}{\numberline {9.1.1}\leavevmode {\color {Chapter }FactorizationsElementWRTNumericalSemigroup}}{39}{subsection.9.1.1}
\contentsline {subsection}{\numberline {9.1.2}\leavevmode {\color {Chapter }LengthsOfFactorizationsElementWRTNumericalSemigroup}}{40}{subsection.9.1.2}
\contentsline {subsection}{\numberline {9.1.3}\leavevmode {\color {Chapter }ElasticityOfFactorizationsElementWRTNumericalSemigroup}}{40}{subsection.9.1.3}
\contentsline {subsection}{\numberline {9.1.4}\leavevmode {\color {Chapter }ElasticityOfNumericalSemigroup}}{40}{subsection.9.1.4}
\contentsline {subsection}{\numberline {9.1.5}\leavevmode {\color {Chapter }DeltaSetOfFactorizationsElementWRTNumericalSemigroup}}{40}{subsection.9.1.5}
\contentsline {subsection}{\numberline {9.1.6}\leavevmode {\color {Chapter }MaximumDegreeOfElementWRTNumericalSemigroup}}{41}{subsection.9.1.6}
\contentsline {subsection}{\numberline {9.1.7}\leavevmode {\color {Chapter }CatenaryDegreeNumericalSemigroup}}{41}{subsection.9.1.7}
\contentsline {subsection}{\numberline {9.1.8}\leavevmode {\color {Chapter }TameDegreeNumericalSemigroup}}{41}{subsection.9.1.8}
\contentsline {chapter}{\numberline {A}\leavevmode {\color {Chapter }Generalities}}{42}{appendix.A}
\contentsline {section}{\numberline {A.1}\leavevmode {\color {Chapter }B\'ezout sequences}}{42}{section.A.1}
\contentsline {subsection}{\numberline {A.1.1}\leavevmode {\color {Chapter }BezoutSequence}}{42}{subsection.A.1.1}
\contentsline {subsection}{\numberline {A.1.2}\leavevmode {\color {Chapter }IsBezoutSequence}}{42}{subsection.A.1.2}
\contentsline {subsection}{\numberline {A.1.3}\leavevmode {\color {Chapter }CeilingOfRational}}{43}{subsection.A.1.3}
\contentsline {section}{\numberline {A.2}\leavevmode {\color {Chapter }Periodic subadditive functions}}{43}{section.A.2}
\contentsline {subsection}{\numberline {A.2.1}\leavevmode {\color {Chapter }RepresentsPeriodicSubAdditiveFunction}}{43}{subsection.A.2.1}
\contentsline {chapter}{\numberline {B}\leavevmode {\color {Chapter }Random functions}}{44}{appendix.B}
\contentsline {section}{\numberline {B.1}\leavevmode {\color {Chapter }Random functions}}{44}{section.B.1}
\contentsline {subsection}{\numberline {B.1.1}\leavevmode {\color {Chapter }RandomNumericalSemigroup}}{44}{subsection.B.1.1}
\contentsline {subsection}{\numberline {B.1.2}\leavevmode {\color {Chapter }RandomListForNS}}{44}{subsection.B.1.2}
\contentsline {subsection}{\numberline {B.1.3}\leavevmode {\color {Chapter }RandomModularNumericalSemigroup}}{44}{subsection.B.1.3}
\contentsline {subsection}{\numberline {B.1.4}\leavevmode {\color {Chapter }RandomProportionallyModularNumericalSemigroup}}{45}{subsection.B.1.4}
\contentsline {subsection}{\numberline {B.1.5}\leavevmode {\color {Chapter }RandomListRepresentingSubAdditiveFunction}}{45}{subsection.B.1.5}
\contentsline {chapter}{\numberline {C}\leavevmode {\color {Chapter }A graphical interface}}{46}{appendix.C}
\contentsline {section}{\numberline {C.1}\leavevmode {\color {Chapter }Graphical interface}}{46}{section.C.1}
\contentsline {subsection}{\numberline {C.1.1}\leavevmode {\color {Chapter }XNumericalSemigroup}}{46}{subsection.C.1.1}
