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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
<div class="pcenter">
<h1>Numerical Semigroups</h1>
<p>( Version 0.96
)</p>
</div>
<p><b> Manuel Delgado
</b>
<br />Email: <span class="URL"><a href="mailto:mdelgado@fc.up.pt">mdelgado@fc.up.pt</a></span>
<br />Homepage: <span class="URL"><a href="http://www.fc.up.pt/cmup/mdelgado">http://www.fc.up.pt/cmup/mdelgado</a></span>
</p><p><b> Pedro A. García-Sánchez
</b>
<br />Email: <span class="URL"><a href="mailto:pedro@ugr.es">pedro@ugr.es</a></span>
<br />Homepage: <span class="URL"><a href="http://www.ugr.es/~pedro">http://www.ugr.es/~pedro</a></span>
</p><p><b> José João Morais
</b>
<br />Email: <span class="URL"><a href="mailto:josejoao@fc.up.pt">josejoao@fc.up.pt</a></span>
</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2005 by Manuel Delgado, Pedro A. García-Sánchez and José João Morais</p>
<p>We adopt the copyright regulations of <strong class="pkg">GAP</strong> as detailed in the copyright notice in the <strong class="pkg">GAP</strong> manual.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>The first author's work was (partially) supported by the <em>Centro de Matemática da Universidade do Porto</em> (CMUP), financed by FCT (Portugal) through the programmes POCTI (Programa Operacional "Ciência, Tecnologia, Inovação") and POSI (Programa Operacional Sociedade da Informação), with national and European Community structural funds and a sabbatical grant of FCT.</p>
<p>The second author was supported by the project MTM2004-01446 and FEDER founds.</p>
<p>The third author acknowledges financial support of FCT and the POCTI program through a scholarship given by <em>Centro de Matemática da Universidade do Porto</em>.</p>
<p>The authors whish to thank J. I. García-García for many helpfull discussions and for helping in the programming of preliminary versions of some functions.</p>
<p><a id="X7982162280BC7A61" name="X7982162280BC7A61"></a></p>
<h3>Colophon</h3>
<p>This work started when the first author visited the University of Granada in part of a sabbatical year. Bug reports, suggestions and comments are, of course, welcome. Please use our email addresses to this effect.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">
Introduction
</span></a>
</div>
<div class="ContChap"><a href="chap2.html#X8324E5D97DC2A801">2 <span class="Heading">
Numerical Semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7E89D7EB7FCC2197">2.1 <span class="Heading">
Generating Numerical Semigroups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86DEEBFE854B60A6">2.1-1 NumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87206D597873EAFF">2.1-2 ModularNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X879171CD7AC80BB5">2.1-3 ProportionallyModularNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79B38AB3816BF3C5">2.1-4 NumericalSemigroupByGenerators</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7EF4254C81ED6665">2.2 <span class="Heading">Some basic tests</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7B1B6B8C82BD7084">2.2-1 IsNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87B02A9F7AF90CB9">2.2-2 RepresentsSmallElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X78906CCD7BEE0E58">2.2-3 RepresentsGapsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84A611557B5ACF42">2.2-4 IsAperyListOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86D5B3517AF376D4">2.2-5 IsSubsemigroupOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X864C2D8E80DD6D16">2.2-6 BelongsToNumericalSemigroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X7A9D13C778697F6C">3 <span class="Heading">
Basic operations with numerical semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7F15FEA980D637AF">3.1 <span class="Heading">
The definitions
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X842721EF8145C3D3">3.1-1 MultiplicityOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A3177E779C49D13">3.1-2 GeneratorsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79D6E4727B612B54">3.1-3 SmallElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F0EDFA77F929120">3.1-4 FirstElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86C699F2800F9ED0">3.1-5 AperyListOfNumericalSemigroupWRTElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X85C89A6B81A07061">3.1-6 DrawAperyListOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8022CC477E9BF678">3.1-7 AperyListOfNumericalSemigroupAsGraph</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X7F97E0127A1D2835">3.2 <span class="Heading">Frobenius Number</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7828CD2B83E380FA">3.2-1 FrobeniusNumberOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B7DFAB2834B2B36">3.2-2 FrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X829A9C517A83FFD5">3.2-3 PseudoFrobeniusOfNumericalSemigroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X787FE6F180C6291F">3.3 <span class="Heading">Gaps</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B1BEC5786C66244">3.3-1 GapsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X81EC33E978C73FEA">3.3-2 FundamentalGapsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7E373952877AC30A">3.3-3 SpecialGapsOfNumericalSemigroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X7969F7F27AAF0BF1">4 <span class="Heading">
Presentations of Numerical Semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7969F7F27AAF0BF1">4.1 <span class="Heading">Presentations of Numerical Semigroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8028208F84950722">4.1-1 FortenTruncatedNCForNumericalSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D8199858781EB41">4.1-2 MinimalPresentationOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81CC5A6C870377E1">4.1-3 GraphAssociatedToElementInNumericalSemigroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap5.html#X8148F05A830EE2D5">5 <span class="Heading">
Constructing numerical semigroups from others
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X782F3AB97ACF84B8">5.1 <span class="Heading">
Adding and removing elements of a numerical semigroup
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7C94611F7DD9E742">5.1-1 RemoveMinimalGeneratorFromNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X865EA8377D632F53">5.1-2 AddSpecialGapOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X874494928728023E">5.1-3 IntersectionOfNumericalSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X83CCE63C82F34C25">5.1-4 QuotientOfNumericalSemigroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X867D9A9A87CEB869">5.2 <span class="Heading">
Constructing the set of all numerical semigroups containing a given numerical semigroup
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7EEA97AB85FE078C">5.2-1 OverSemigroupsNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X87369D567AA6DBA0">5.2-2 NumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X86970F6A868DEA95">5.2-3 NumericalSemigroupsWithGenus</a></span>
</div>
</div>
<div class="ContChap"><a href="chap6.html#X83C597EC7FAA1C0F">6 <span class="Heading">
Irreducible numerical semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X83C597EC7FAA1C0F">6.1 <span class="Heading">
Irreducible numerical semigroups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87D62468791EDE8A">6.1-1 IsIrreducibleNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7BCDAFE3791A3C48">6.1-2 IsSymmetricNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X84125DC485D48A88">6.1-3 IsPseudoSymmetricNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C8AB03F7E0B71F0">6.1-4 AnIrreducibleNumericalSemigroupWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78345A267ADEFBAB">6.1-5 IrreducibleNumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8227EF2B7F67E2FB">6.1-6 DecomposeIntoIrreducibles</a></span>
</div>
</div>
<div class="ContChap"><a href="chap7.html#X83C2F0CF825B3869">7 <span class="Heading">
Ideals of numerical semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X83C2F0CF825B3869">7.1 <span class="Heading">
Ideals of numerical semigroups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X78E5F44E81485C17">7.1-1 IdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X85BD6FAD7EA3B5DD">7.1-2 IsIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7CF1EA687B137C3D">7.1-3 MinimalGeneratingSystemOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X86A9200283D1B32B">7.1-4 GeneratorsOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X81E445518529C175">7.1-5 AmbientNumericalSemigroupOfIdeal</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7BDEBFCB7A1DAFC7">7.1-6 SmallElementsOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X87508E7A7CFB0B20">7.1-7 BelongsToIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7B39610D7AD5A654">7.1-8 SumIdealsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X857FE5C57EE98F5E">7.1-9 MultipleOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X78743CE2845B5860">7.1-10 SubtractIdealsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7C2DAB737ECE7D34">7.1-11 DifferenceOfIdealsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X803921F97BEDCA88">7.1-12 TranslationOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X82156F18807B00BF">7.1-13 HilbertFunctionOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7C00A86F83024003">7.1-14 BlowUpIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X82C2BF5E840C815D">7.1-15 ReductionNumberIdealNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X829EACA378BE3665">7.1-16 MaximalIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X84B5121C7EEECB30">7.1-17 BlowUpOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7B29CDA7783FC0D2">7.1-18 MicroInvariantsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7876199778D6B320">7.1-19 IsGradedAssociatedRingNumericalSemigroupCM</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X835890A078F5D6DC">7.1-20 CanonicalIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X85CC100F78608D5E">7.1-21 IntersectionIdealsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7.html#X7A04B8887F493733">7.1-22 IsMonomialNumericalSemigroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap8.html#X7D2E70FC82D979D3">8 <span class="Heading">
Numerical semigroups with maximal embedding dimension
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X7D2E70FC82D979D3">8.1 <span class="Heading">
Numerical semigroups with maximal embedding dimension
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X867615F8846824EB">8.1-1 IsMEDNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X86C8358D8530106F">8.1-2 MEDNumericalSemigroupClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X848FD3FA7DB2DD4C">8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap8.html#X82E40EFD83A4A186">8.2 <span class="Heading">
Numerical semigroups with the Arf property and Arf closures
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X8255C5907F8968B9">8.2-1 IsArfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7FE10E2F85CB01A2">8.2-2 ArfNumericalSemigroupClosure</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap8.html#X7C0D2F7986165DDE">8.2-3 MinimalArfGeneratingSystemOfArfNumericalSemigroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap9.html#X7C76B03A84BA7574">9 <span class="Heading">
Catenary and Tame degrees of numerical semigroups
</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap9.html#X7FDB54217B15148F">9.1 <span class="Heading">
Factorizations in Numerical Semigroups
</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X78C6D3BF7C7C2760">9.1-1 FactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7FDE4F94870951B1">9.1-2 LengthsOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X85C2987C7827D18D">9.1-3 ElasticityOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7FE2D6F77BE96716">9.1-4 ElasticityOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X8384DBFE7D82A634">9.1-5 DeltaSetOfFactorizationsElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7E3ED34D78F3A8CA">9.1-6 MaximumDegreeOfElementWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X7FAF204E85D9C21B">9.1-7 CatenaryDegreeOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X787334508257C510">9.1-8 CatenaryDegreeOfElementNS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap9.html#X860BDF5B85975B73">9.1-9 TameDegreeOfNumericalSemigroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chapA.html#X7AF8D94A7E56C049">A <span class="Heading">Generalities</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chapA.html#X7A5D608487A8C98F">A.1 <span class="Heading">Bézout sequences</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X86859C84858ECAF1">A.1-1 BezoutSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X86C990AC7F40E8D0">A.1-2 IsBezoutSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7C9DCBAF825CF7B2">A.1-3 CeilingOfRational</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chapA.html#X7D3D347987953F44">A.2 <span class="Heading">Periodic subadditive functions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X8466A4DC82F07579">A.2-1 RepresentsPeriodicSubAdditiveFunction</a></span>
</div>
</div>
<div class="ContChap"><a href="chapB.html#X86746B487B54A2D6">B <span class="Heading">Random functions</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chapB.html#X86746B487B54A2D6">B.1 <span class="Heading">Random functions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X7CC477867B00AD13">B.1-1 RandomNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X79E73F8787741190">B.1-2 RandomListForNS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X82E22E9B843DF70F">B.1-3 RandomModularNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X8598F10A7CD4A135">B.1-4 RandomProportionallyModularNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapB.html#X8665F6B08036AFFB">B.1-5 RandomListRepresentingSubAdditiveFunction</a></span>
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