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<div class="pcenter" >
<h1><strong class="pkg" >numericalsgps</strong>-- a package for numerical semigroups</h1>
<p> Version 1 .4 .0 </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>
</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1" ></a></p>
<h3>Copyright</h3>
<p>© 2005 --2015 Centro de Matemática da Universidade do Porto, Portugal and Universidad de Granada, Spain</p>
<p><em>Numericalsgps</em> is free software; you can redistribute it and/or modify it under the terms of the <span class="URL" ><a href="http://www.fsf.org/licenses/gpl.html " >GNU General Public License</a></span> as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. For details, see the file 'GPL' included in the package or see the FSF's own site.</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA" ></a></p>
<h3>Acknowledgements</h3>
<p>The authors wish to thank the contributors of the package. A full list with the help received is available in Appendix <a href="chapC.html#X7F1146137C92FF0E"><span class="RefLink">C</span></a>. We are also in debt with H. Schönemann, C. Söeger and M. Barakat for their fruitful advices concerning SingularInterface, Singular, Normaliz, NormalizInterface and GradedModules.</p>
<p>The maintainers want to thank the organizers of <code class="code" >GAPDays</code> in their several editions.</p>
<p>The authors also thank the Centro de Servicios de Informática y Redes de Comunicaciones (CSIRC), Universidad de Granada, for providing the computing time, specially Rafael Arco Arredondo for installing this package and the extra software needed in alhambra.ugr.es, and Santiago Melchor Ferrer for helping in job submission to the cluster.</p>
<p>The first and second authors warmly thank María Burgos for her support and help.</p>
<p><strong class="button" >Funding</strong></p>
<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 programs 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 projects MTM2004-01446 and MTM2007-62346 , the Junta de Andalucía group FQM-343 , 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 first author was (partially) supported by the FCT project PTDC/MAT/65481 /2006 and also by the <em>Centro de Matemática da Universidade do Porto</em> (CMUP), funded by the European Regional Development Fund
<p>Both maintainers were (partially) supported by the projects MTM2010-15595 and MTM2014-55367 -P, which were funded by Ministerio de Economía y Competitividad and the Fondo Europeo de Desarrollo Regional FEDER.</p>
<p>Both maintainers want to acknowledge partial support by CMUP (UID/MAT/00144 /2013 and UID/MAT/00144 /2019 ), which is funded by FCT (Portugal) with national (MEC) and European structural funds through the programs FEDER, under the partnership agreement PT2020.</p>
<p>Both maintainers were also partially supported by the project MTM2017-84890 -P, which is funded by Ministerio de Economía y Competitividad and Fondo Europeo de Desarrollo Regional FEDER.</p>
<p>The first author acknowledges a sabbatical grant from the FCT: SFRH/BSAB/142918 /2018 .</p>
<p>The second author was supported in part by Grant PGC2018-096446 -B-C21 funded by MCIN/AEI/10 .13039 /501100011033 and by "ERDF A way of making Europe" .</p>
<p>Both maintainers were partially supported by CMUP, member of LASI, which is financed by Portuguese national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project with reference UIDB/00144 /2020 and UIDP/00144 /2020 .</p>
<p>Both maintainers acknowledge the "Proyecto de Excelencia de la Junta de Andalucía" (ProyExcel 00868 ).</p>
<p><a id="X7982162280BC7A61" name="X7982162280BC7A61" ></a></p>
<h3>Colophon</h3>
<p>This work started when (in 2004 ) the first author visited the University of Granada in part of a sabbatical year. Since Version 0 .96 (released in 2008 ), the package is maintained by the first two authors. Bug reports, suggestions and comments are, of course, welcome. Please use our email addresses to this effect.</p>
<p>If you have benefited from the use of the numerigalsgps GAP package in your research, please cite it in addition to GAP itself, following the scheme proposed in <span class="URL" ><a href="https://www.gap-system.org/Contacts/cite.html " >https://www.gap-system.org/Contacts/cite.html</a></span>.</p>
<p>If you have predominantly used the functions in the Appendix, contributed by other authors, please cite in addition these authors, referring "software implementations available in the GAP package NumericalSgps" .</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8" ></a></p>
<div class="contents" >
<h3>Contents<a id="contents" name="contents" ></a></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="tocline" ><span class="nocss" > </span><a href="chap2.html#X7E89D7EB7FCC2197">2.1 <span class="Heading">
Generating Numerical Semigroups
</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X7D74299B8083E882">2.1-1 NumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X86D9D2EE7E1C16C2">2.1-2 NumericalSemigroupBySubAdditiveFunction</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X799AC8727DB61A99">2.1-3 NumericalSemigroupByAperyList</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X81A7E3527998A74A">2.1-4 NumericalSemigroupBySmallElements</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X7BB0343D86EC5FEC">2.1-5 NumericalSemigroupByGaps</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X86AC8B0E7C11147F">2.1-6 NumericalSemigroupByFundamentalGaps</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X7ACD94F478992185">2.1-7 NumericalSemigroupByAffineMap</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X87206D597873EAFF">2.1-8 ModularNumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X879171CD7AC80BB5">2.1-9 ProportionallyModularNumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X7D8F9D2A8173EF32">2.1-10 NumericalSemigroupByInterval</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X7C800FB37D76612F">2.1-11 NumericalSemigroupByOpenInterval</a></span>
</div></div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chap2.html#X7EF4254C81ED6665">2.2 <span class="Heading">Some basic tests</span></a>
</span>
<div class="ContSSBlock" >
<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#X79CA175481F8105F">2.2-6 IsSubset</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap2.html#X864C2D8E80DD6D16">2.2-7 BelongsToNumericalSemigroup</a></span>
</div></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="tocline" ><span class="nocss" > </span><a href="chap3.html#X87AF9D4F7FD9E820">3.1 <span class="Heading">
Invariants
</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X80D23F08850A8ABD">3.1-1 Multiplicity</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X850F430A8284DF9A">3.1-2 Generators</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7884AE27790E687F">3.1-3 EmbeddingDimension</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X84A6B16E8113167B">3.1-4 SmallElements</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7A56569F853DADED">3.1-5 Length</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7F0EDFA77F929120">3.1-6 FirstElementsOfNumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7D2B3AA9823371AE">3.1-7 ElementsUpTo</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X81A2505E8120F4D7"><code>3.1-8 <span>\</span>[ <span>\</span>]</code></a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7A34F16F8112C2B5"><code>3.1-9 \{ \}</code></a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X84345D5E7CAA9B77">3.1-10 NextElementOfNumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7B6C82DD86E5422F">3.1-11 ElementNumber_NumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X78F4A7A7797E26D4">3.1-12 NumberElement_NumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X867ABF7C7991ED7C">3.1-13 Iterator</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7E6F5D6F7B0C9635">3.1-14 Difference</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7CB24F5E84793BE1">3.1-15 AperyList</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X80431F487C71D67B">3.1-16 AperyList</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7D06B00D7C305C64">3.1-17 AperyList</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X8022CC477E9BF678">3.1-18 AperyListOfNumericalSemigroupAsGraph</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X80B398537887FD87">3.1-19 KunzCoordinates</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7C21E5417A3894EC">3.1-20 KunzPolytope</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7802096584D32795">3.1-21 CocycleOfNumericalSemigroupWRTElement</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X847BAD9480D186C0">3.1-22 FrobeniusNumber</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X835C729D7D8B1B36">3.1-23 Conductor</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X861DED207A2B5419">3.1-24 PseudoFrobenius</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X865E2E12804CFCD3">3.1-25 Type</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X8688B1837E4BC079">3.1-26 Gaps</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7F71983880DF4B9D">3.1-27 Weight</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7EB81BF886DDA29A">3.1-28 Deserts</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X82B1868F7A780B49">3.1-29 IsOrdinary</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X83D4AFE882A79096">3.1-30 IsAcute</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7CCFC5267FD27DDE">3.1-31 Holes</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X794E615F85C2AAB0">3.1-32 LatticePathAssociatedToNumericalSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7E9C8E157C4EAAB0">3.1-33 Genus</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X7EC438CC7BF539D0">3.1-34 FundamentalGaps</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap3.html#X803D550C78717A7C">3.1-35 SpecialGaps</a></span>
</div></div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap3.html#X7EE22CA979CCAAB9" >3 .2 <span class="Heading" >Wilf's conjecture</span></a>
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X78C2F4C77FB096F0" >3 .2 -1 WilfNumber</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X80F9EC9A7BF4E606" >3 .2 -2 EliahouNumber</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7B45623E7D539CB6" >3 .2 -3 ProfileOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap3.html#X7846F90E7EA43C47" >3 .2 -4 EliahouSlicesOfNumericalSemigroup</a ></span >
</div ></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="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7969F7F27AAF0BF1" >4 .1 <span class="Heading" >Presentations of Numerical Semigroups</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81A2C4317A0BA48D" >4 .1 -1 MinimalPresentation</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81CC5A6C870377E1" >4 .1 -2 GraphAssociatedToElementInNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X815C0AF17A371E3E" >4 .1 -3 BettiElements</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7FC66A1B82E86FAF" >4 .1 -4 IsMinimalRelationOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8750A6837EF75CA2" >4 .1 -5 AllMinimalRelationsOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7A9B5AE782CAEA2F" >4 .1 -6 DegreesOfPrimitiveElementsOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7C42DEB68285F2B8" >4 .1 -7 ShadedSetOfElementInNumericalSemigroup</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X795E7F5682A6C8B3" >4 .2 <span class="Heading" >Binomial ideals associated to numerical semigroups</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7E6BBAA7803DE7F3" >4 .2 -1 BinomialIdealOfNumericalSemigroup</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7D7EA20F818A5994" >4 .3 <span class="Heading" >Uniquely Presented Numerical Semigroups</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7C6F554486274CAE" >4 .3 -1 IsUniquelyPresented</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X79C010537C838154" >4 .3 -2 IsGeneric</a ></span >
</div ></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="tocline" ><span class=="" > <span ><a href="chap3.html#X7EE22CA979CCAAB9" >3 .2 <span class="Heading" >Wilf's conjecture</span></a>
Adding and removing elements of <pan class"><br >span class" nocss"&;/> href=chap3.htmlX78C2F4C77FB096F0>.- </a<span>
</span ></a >
<span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span <span class="ContSS"><br /><span classnocss>nbsp </span ><a href="chap3.html#X7B45623E7D539CB6" >3 .2 -3 ProfileOfNumericalSemigroup</a ></span >
<">5.1-2 AddSpecialGapOfNumericalSemigroup<a<spanjava.lang.StringIndexOutOfBoundsException: Index 155 out of bounds for length 155
</div </java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<div class="ContSect" ><span class="tocline" ><span class= Presentations ofNumerical Semigroups
</span >
<">
<</spa
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<span class="ContSS" ><br /><span class="< =ContSS>br /><span class=" nocss"&&;/< " chap4.#X81A2C4317A0BA48D>.-MinimalPresentation<a />
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Irreducible numerical semigroups
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap6.html#X83C597EC7FAA1C0F" >6 .1 <span class="Heading" >
Irreducible numerical semigroups
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X83E8CC8F862D1FC0" >6 .1 -1 IsIrreducible</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X7C381E277917B0ED" >6 .1 -2 IsSymmetric</a ></span >
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<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 >
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap6.html#X8784D11578C912F2" >6 .2 -10 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber</a ></span >
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap7.html#X81CD9B12807EEA85" >7 .3 <span class="Heading" >
Blow ups and closures
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap7.html#X82156F18807B00BF" >7 .3 -1 HilbertFunctionOfIdealOfNumericalSemigroup</a ></span >
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Patterns for ideals
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X85CD144384FD55F3" >8 .2 -4 ArfNumericalSemigroupsWithFrobeniusNumber</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7E308CCF87448182" >8 .2 -5 ArfNumericalSemigroupsWithFrobeniusNumberUpTo</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X80A13F7C81463AE5" >8 .2 -6 ArfNumericalSemigroupsWithGenus</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X80EB35C17C83694D" >8 .2 -7 ArfNumericalSemigroupsWithGenusUpTo</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7EE73B2F813F7E85" >8 .2 -8 ArfNumericalSemigroupsWithGenusAndFrobeniusNumber</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7CC73F15831B06CE" >8 .2 -9 ArfSpecialGaps</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7DD2831683F870C5" >8 .2 -10 ArfOverSemigroups</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X8052BCE67CC2472F" >8 .2 -11 IsArfIrreducible</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X848E5559867D2D81" >8 .2 -12 DecomposeIntoArfIrreducibles</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap8.html#X7E6D857179E5BF1B" >8 .3 <span class="Heading" >
Saturated numerical semigroups
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X81CCD9A88127E549" >8 .3 -1 IsSaturated</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X78E6F00287A23FC1" >8 .3 -2 SaturatedClosure</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap8.html#X7CC07D997880E298" >8 .3 -3 SaturatedNumericalSemigroupsWithFrobeniusNumber</a ></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap9.html#X7B6F914879CD505F" >9 <span class="Heading" >
Nonunique invariants for factorizations in numerical semigroups
</span ></a >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a <ouse- items-whatever suits .
Factorizations in Numerical Semigroups
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<class"ContSS> >spanclass=" nocss"nbsp&;span>ahref=" =".htmlXEF105B82447F30" >9 1 - Factorizations</>/>
<span consideritmore usefultopermit proprietary withthe
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7C5EED6D852C24DD" >9 .1 -5 LShapes</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#html#X7F8B10C2870932B8I/<span
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< "ontSect" <span class=tocline"<span =" "> <span><ahref=" hap9hX846FEE457D4EC03D">9.2 <span class=" Heading">
Invariants based on lengths
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7D4CC092859AF81F" >9 .2 -1 LengthsOfFactorizationsIntegerWRTList</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7FDE4F94870951B1" >9 .2 -2 LengthsOfFactorizationsElementWRTNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X860E461182B0C6F5" >9 .2 -3 Elasticity</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7A2B01BB87086283" >9 .2 -4 Elasticity</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X79C953B5846F7057" >9 .2 -5 DeltaSet</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DB8BA5B7D6F81CB" >9 .2 -6 DeltaSet</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7A08CF05821DD2FC" >9 .2 -7 DeltaSetPeriodicityBoundForNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8123FC0E83ADEE45" >9 .2 -8 DeltaSetPeriodicityStartForNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X80B5DF908246BEB1" >9 .2 -9 DeltaSetListUpToElementWRTNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X85C6973E81583E8B" >9 .2 -10 DeltaSetUnionUpToElementWRTNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X83B06062784E0FD9" >9 .2 -11 DeltaSet</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7AEFE27E87F51114" >9 .2 -12 MaximumDegree</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7F8B10C2870932B8" >9 .2 -13 IsAdditiveNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X790308B07AB1A5C8" >9 .2 -14 MaximalDenumerant</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DFC4ED0827761C1" >9 .2 -15 MaximalDenumerantOfSetOfFactorizations</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X811E5FFB83CCA4CE" >9 .2 -16 MaximalDenumerant</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X87F633D98003DE52" >9 .2 -17 Adjustment</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X84F5CA8D7B0F6C02" >9 .3 <span class="Heading" >
Invariants based on distances
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X86F9D7868100F6F9" >9 .3 -1 CatenaryDegree</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DDB40BB84FF0042" >9 .3 -2 AdjacentCatenaryDegreeOfSetOfFactorizations</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X86E0CAD28655839C" >9 .3 -3 EqualCatenaryDegreeOfSetOfFactorizations</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X845D850F7812E176" >9 .3 -4 MonotoneCatenaryDegreeOfSetOfFactorizations</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X797147AA796D1AFE" >9 .3 -5 CatenaryDegree</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X80D478418403E7CB" >9 .3 -6 TameDegree</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X785B83F17BEEA894" >9 .3 -7 CatenaryDegree</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X863E3EF986764267" >9 .3 -8 DegreesOffEqualPrimitiveElementsOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X780E2C737FA8B2A9" >9 .3 -9 EqualCatenaryDegreeOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7E19683D7ADDE890" >9 .3 -10 DegreesOfMonotonePrimitiveElementsOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7E0458187956C395" >9 .3 -11 MonotoneCatenaryDegreeOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X809D97A179765EE6" >9 .3 -12 TameDegree</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7F7619BD79009B64" >9 .3 -13 TameDegree</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X78EBC6A57B8167E6" >9 .4 <span class="Heading" >
Primality
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X83075D7F837ACCB8" >9 .4 -1 OmegaPrimality</a ></span >
<span class="ContSS" ><br /><span class="div class="ContChap"><a href="chap10.html#X7D2C77607815273E" ContChap"><a=chap10.#X7D2C77607815273E" > span class="Heading" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X80B48B7886A93FAC" >9 .4 -3 OmegaPrimality</a ></span >
</div ></div >
<div class"ContSect><span class=" tocline"><span class=" nocss"> </span><a href=" chap9.html #X86735EEA780CECDA">9.5 <span class=" Heading">
Homogenization of Numerical Semigroups
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > <
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X85D03DBB7BA3B1FB" >9 .5 -2 FactorizationsInHomogenizationOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X857CC7FF85C05318" >9 .5 -3 HomogeneousBettiElementsOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DFFCAC87B3B632B" >9 .5 -4 HomogeneousCatenaryDegreeOfNumericalSemigroup</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X7A54E9FD7D4CB18F" >9 .6 <span class="Heading" >
Divisors, posets
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X853930E97F7F8A43" >9 .6 -1 MoebiusFunctionAssociatedToNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7DF6825185C619AC" >9 .6 -2 MoebiusFunction</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8771F39A7C7E031E" >9 .6 -3 DivisorsOfElementInNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X871CD69180783663" >9 .6 -4 NumericalSemigroupByNuSequence</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7F4CBFF17BBB37DE" >9 .6 -5 NumericalSemigroupByTauSequence</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X82D8A59083FCDF46" >9 .7 <span class="Heading" >
Feng-Rao distances and numbers
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7939BCE08655B62D" >9 .7 -1 FengRaoDistance</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X83F9F4C67D4535EF" >9 .7 -2 FengRaoNumber</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap9.html#X79A8A15087CEE8C1" >9 .8 <span class="Heading" >
Numerical semigroups with Apéry sets having special factorization properties
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X7B894ED27D38E4B5" >9 .8 -1 IsPure</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8400FB5D81EFB5FE" >9 .8 -2 IsMpure</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X80B707EE79990E1E" >9 .8 -3 IsHomogeneousNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap9.html#X8630DEF77A350D76" >9 .8 -4 IsSuperSymmetricNumericalSemigroup</a ></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap10.html#X7D2C77607815273E" >10 <span class="Heading" >
Polynomials and numerical semigroups
</span ></a >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X808FAEE28572191C" >10 .1 <span class="Heading" >
Generating functions or Hilbert series
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X8391C8E782FBFA8A" >10 .1 -1 NumericalSemigroupPolynomial</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7F59E1167C1EE578" >10 .1 -2 IsNumericalSemigroupPolynomial</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X855497F77D13436F" >10 .1 -3 NumericalSemigroupFromNumericalSemigroupPolynomial</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X780479F978D166B0" >10 .1 -4 HilbertSeriesOfNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X87C88E5C7B56931F" >10 .1 -5 GraeffePolynomial</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X87A46B53815B158F" >10 .1 -6 IsCyclotomicPolynomial</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7D9618ED83776B0B" >10 .1 -7 IsKroneckerPolynomial</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X8366BB727C496D31" >10 .1 -8 IsCyclotomicNumericalSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7B428FA2877EC733" >10 .1 -9 CyclotomicExponentSequence</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7A33BA9B813A4070" >10 .1 -10 WittCoefficients</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X82C6355287C3BDD1" >10 .1 -11 IsSelfReciprocalUnivariatePolynomial</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X7EEF2A1781432A2D" >10 .2 <span class="Heading" >
Semigroup of values of algebraic curves
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7FFF949A7BEEA912" >10 .2 -1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X834D6B1A7C421B9F" >10 .2 -2 IsDeltaSequence</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X824ABFD680A34495" >10 .2 -3 DeltaSequencesWithFrobeniusNumber</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X87B819B886CA5A5C" >10 .2 -4 CurveAssociatedToDeltaSequence</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7E2C3E9A7DE7A078" >10 .2 -5 SemigroupOfValuesOfPlaneCurve</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7F88774F7812D30E" >10 .2 -6 SemigroupOfValuesOfCurve_Local</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X8597259279D1E793" >10 .2 -7 SemigroupOfValuesOfCurve_Global</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7EE8528484642CEE" >10 .2 -8 GeneratorsModule_Global</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X836D31F787641C22" >10 .2 -9 GeneratorsKahlerDifferentials</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7A04B8887F493733" >10 .2 -10 IsMonomialNumericalSemigroup</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap10.html#X84C670E1826F8B92" >10 .3 <span class="Heading" >
Semigroups and Legendrian curves
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap10.html#X7980A7CE79F09A89" >10 .3 -1 LegendrianGenericNumericalSemigroup</a ></span >
</div ></div >
</div >
<div class="ContChap" ><a href="chap11.html#X7D92A1997D098A00" >11 <span class="Heading" >
Affine semigroups
</span ></a >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7E39DA7780D02DF5" >11 .1 <span class="Heading" >
Defining affine semigroups
</span ></a >
</span >
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7D7B03E17C8DBEA2" >11 .1 -1 AffineSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X855C8667830AEDDC" >11 .1 -2 AffineSemigroupByEquations</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7846AD1081C14EF1" >11 .1 -3 AffineSemigroupByInequalities</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7CC110D4798AAD99" >11 .1 -4 AffineSemigroupByPMInequality</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X83F6DDB787E07771" >11 .1 -5 AffineSemigroupByGaps</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7A3648D67CF81370" >11 .1 -6 FiniteComplementIdealExtension</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X8361194C86AE807B" >11 .1 -7 Gaps</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X867B27BD81104BEE" >11 .1 -8 Genus</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X80C3CD2082CE02F7" >11 .1 -9 PseudoFrobenius</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X82D42FCE81F20277" >11 .1 -10 SpecialGaps</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X84FDF85D7CDEDF3E" >11 .1 -11 Generators</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7ED1549486C251CA" >11 .1 -12 MinimalGenerators</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X80516BCC78FDD45D" >11 .1 -13 RemoveMinimalGeneratorFromAffineSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7B78E02F7C50583F" >11 .1 -14 AddSpecialGapOfAffineSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X844806D97B4781B5" >11 .1 -15 AsAffineSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7A2902207BAA3936" >11 .1 -16 IsAffineSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X851788D781A13C50" >11 .1 -17 BelongsToAffineSemigroup</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X8607B621833FAECB" >11 .1 -18 IsFull</a ></span >
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7A1CE5A98425CEA1" >11 .1 -22 BasisOfGroupGivenByEquations</a ></span >
</div ></div >
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X7F13DF9D7A4FB547"
Gluingsspan =ContSS"><br ><class" "nbspn</>a href=" hap12html X82A8863E78650FC412 .- <aspan
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap11.html#X86A1018D7CB7BA81" >11 .3 <span class="Headingspan class" "<br >span =" nocss">nbsp; </span><a href=" chap12.html #X7856241678224958">12.1-5 GoodSemigroup</a></span>
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< class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X795EEE4481E0497C" >11 .3 -1 CircuitsOfKernelCongruence</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X78B04C198258D3F8" >11 .3 -2 PrimitiveRelationsOfKernelCongruence</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7EE005267DEBC1DE" >11 .3 -3 GeneratorsOfKernelCongruence</a ></java.lang.StringIndexOutOfBoundsException: Index 150 out of bounds for length 136
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<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X86BCBD32781EBC2D" >11 .3 -7 BettiElements</a ></span >
<span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap11.html#X7C1B355F83A55285" >11 .3 -8 ShadedSetOfElementInAffineSemigroup</a ></span >
<span class="ContSS" ><
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X79EC6F7583B0CBDD">11.3-10 IsUniquelyPresented</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7DCAFC5F7F74F3CB">11.3-11 DegreesOfPrimitiveElementsOfAffineSemigroup</a></span>
</div></div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chap11.html#X80A934B0826E21A6">11.4 <span class="Heading">
Factorizations in affine semigroups
</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X8780C7E5830B9AE2">11.4-1 FactorizationsVectorWRTList</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X820A0D06857C4EF5">11.4-2 Factorizations</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7F394FA67BE5151B">11.4-3 Elasticity</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X819CDBAA84DB7E83">11.4-4 Elasticity</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X839549448300AD26">11.4-5 DeltaSet</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X80742F2F7DECDB4C">11.4-6 CatenaryDegree</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7EADD306875FCBE6">11.4-7 EqualCatenaryDegreeOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X84FE571A7E9E1AE9">11.4-8 HomogeneousCatenaryDegreeOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X8510C1527F2FE18E">11.4-9 MonotoneCatenaryDegreeOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X8457595E7AA542E6">11.4-10 TameDegree</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X850790EE8442FD7D">11.4-11 OmegaPrimality</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7A3571E187D0FCDE">11.4-12 OmegaPrimality</a></span>
</div></div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chap11.html#X849D1ECC808F2BBA">11.5 <span class="Heading">
Finitely generated ideals of affine semigroups
</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X85775D4E7B9C7DAB">11.5-1 IdealOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X82A647B27FDFE49B">11.5-2 IsIdealOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7F16A5A27CBB7B93">11.5-3 MinimalGenerators</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X8086C1EE7EAAB33D">11.5-4 Generators</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X82D18D7B877582B0">11.5-5 AmbientAffineSemigroupOfIdeal</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7B0FBEC285F54B8D">11.5-6 IsIntegral</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7F00912C853AA83D">11.5-7 BelongsToIdealOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X83A4392281981911">11.5-8 SumIdealsOfAffinSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7D056A0C7F868209">11.5-9 MultipleOfIdealOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X788264A27ACD6AB5">11.5-10 TranslationOfIdealOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7880F29982B559F2">11.5-11 UnionIdealsOfAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X7ED03363783D8FCD">11.5-12 Intersection</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap11.html#X79ECACE4793A6B00">11.5-13 MaximalIdeal</a></span>
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<div class="ContChap" ><a href="chap12.html#X7A9271AC84C7277F">12 <span class="Heading">
Good semigroups
</span></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chap12.html#X82B9F71084D2358E">12.1 <span class="Heading">
Defining good semigroups
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</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X79E86DEE79281BF2">12.1-1 IsGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X82A8863E78650FC4">12.1-2 NumericalSemigroupDuplication</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X873FE7B37A747247">12.1-3 AmalgamationOfNumericalSemigroups</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X855341C57F43DB72">12.1-4 CartesianProductOfNumericalSemigroups</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X7856241678224958">12.1-5 GoodSemigroup</a></span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chap12.html#X8431465B82643392">12.2 <span class="Heading">
Notable elements
</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X79EBBF6D7A2C9A12">12.2-1 BelongsToGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X78A2A60481EE02E7">12.2-2 Conductor</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X7B2F716B7985872B">12.2-3 Multiplicity</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X792BCCF87CF63122">12.2-4 IsLocal</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X836AB83682858A11">12.2-5 SmallElements</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X82D40159783F0D48">12.2-6 RepresentsSmallElementsOfGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X7E538585815C94D0">12.2-7 GoodSemigroupBySmallElements</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X83F444E586D96723">12.2-8 MaximalElementsOfGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X8503AC767A90C2BD">12.2-9 IrreducibleMaximalElementsOfGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X78B456D27856761F">12.2-10 GoodSemigroupByMaximalElements</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X8742875C836C9488">12.2-11 MinimalGoodGenerators</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X806865CB794CAC5D">12.2-12 ProjectionOfAGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X7D70CD958333D49B">12.2-13 Genus</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X81BD57ED80145EB0">12.2-14 Length</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X809E0C077A613806">12.2-15 AperySetOfGoodSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X7B234A537F0C0AEF">12.2-16 StratifiedAperySetOfGoodSemigroup</a></span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </pan>
Symmetric good semigroups
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span=ContSS"< >span =nocss"   /span< hrefhref"hap12html#X85A0D9C485431828">12.3-1 IsSymmetric</a></span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><<class="ContSect" >span class">spanclass=>&bsp<span< href=chapChtml
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<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X87248BD481228F36">12.4-1 ArfClosure</a></span>
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div=ContSect"< class=" tocline>spanclass"nocss>nbsp/span>href" html#X7FA8DCAC7951F7FB">12.5 <span class="Heading">
Good ideals
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<span class="ContSS" ><br /><span class="nocss" > </<spanclass=ContSS"><br /< " nocss>&bsp&;</span>ahref".#X8503AC767A90C2BD">12.2-9 IrreducibleMaximalElementsOfGoodSemigroup</a></span>
><a ".#X7E4FC6DB794992E0">12.5-2 GoodGeneratingSystemOfGoodIdeal</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X82D384397EE5CAC4">12.5-3 AmbientGoodSemigroupOfGoodIdeal</a></span>
<span class<spanclass"" >br/< class"ocss>;nbsp;<>a=chap12html#X8742875C836C9488">12.2-11 MinimalGoodGenerators</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chap12.html#X797999937E4E1E2B">12.5-5 BelongsToGoodIdeal</a></span>
<span class="ContSS" ><br />< class="ContSS" <br /< class">&&bsp<span>href=chap12.#X806865CB794CAC5D" 12 .12 ProjectionOfAGoodSemigroup
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<div class="ContSect" ><span class="tocline<div class" ContSect><span class="" >spanclass=">a" .html#X80A3D64386A152EB">12.4 <span class="Heading">
Using java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
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java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 138
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<div class="ContChap" ><a href="chapA.html#X7AF8D94A7E56C049">A <span class="Heading">Generalities</span></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapA.html#X7A5D608487A8C98F">A.1 <span class="Heading">Bézout sequences</span></a>
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<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>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapA.html#X7D3D347987953F44">A.2 <span class="Heading">Periodic subadditive functions</span></a>
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<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapA.html#X8466A4DC82F07579">A.2-1 RepresentsPeriodicSubAdditiveFunction</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapA.html#X7DFEDA6B87BB2E1F">A.2-2 IsListOfIntegersNS</a></span>
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<div class="ContChap" ><a href="chapB.html#X86746B487B54A2D6">B <span class="Heading">"Random" functions</span></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapB.html#X7F3FF11486C5CA4B">B.1 <span class="Heading">Random functions for numerical semigroups</span></a>
</span>
<div class="ContSSBlock" >
<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>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapB.html#X7B459C8C825194E4">B.1-6 NumericalSemigroupWithRandomElementsAndFrobenius</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapB.html#X78A2A0107CCBBB79">B.1-7 RandomNumericalSemigroupWithGenus</a></span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapB.html#X7D86D133840F6860">B.2 <span class="Heading">Random functions for affine semigroups</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapB.html#X7FBFEE457E823E15">B.2-1 RandomAffineSemigroupWithGenusAndDimension</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapB.html#X82569F0079599515">B.2-2 RandomAffineSemigroup</a></span>
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapB.html#X7F7BB53A7DF77ED5">B.2-3 RandomFullAffineSemigroup</a></span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapB.html#X7DB89F2078A6095F">B.3 <span class="Heading">Random functions for good semigroups</span></a>
</span>
<div class="ContSSBlock" >
<span class="ContSS" ><br /><span class="nocss" > </span><a href="chapB.html#X7F582A997B4B05EE">B.3-1 RandomGoodSemigroupWithFixedMultiplicity</a></span>
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</div>
<div class="ContChap" ><a href="chapC.html#X7F1146137C92FF0E">C <span class="Heading">Contributions</span></a>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X8516272A7ACC7C02">C.1 <span class="Heading">Functions implemented by A. Sammartano</span></a>
</span>
</div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X821A695C7C0BDF59">C.2 <span class="Heading">Functions implemented by C. O'Neill</span></a>
</span>
</div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X7F4C9F8A7F7CDBC8">C.3 <span class="Heading">Functions implemented by K. Stokes</span></a>
</span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X81478D2D862B6213">C.4 <span class="Heading">Functions implemented by I. Ojeda and C. J. Moreno Ávila</span></a>
</span>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X7C7DCFA37C8B5260">C.5 <span class="Heading">Functions implemented by I. Ojeda</span></a>
</span>
</div>
<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X8549AE427919FFDC">C.6 <span class="Heading">Functions implemented by A. Sánchez-R. Navarro</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X7FAE71B27B0E3889">C.7 <span class="Heading">Functions implemented by G. Zito</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X85067C3383705D0B">C.8 <span class="Heading">Functions implemented by A. Herrera-Poyatos</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X81EA8996840BD031">C.9 <span class="Heading">Functions implemented by Benjamin Heredia</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X7ED672F578B6FDC3">C.10 <span class="Heading">Functions implemented by Juan Ignacio García-García</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X8348844883A78B05">C.11 <span class="Heading">Functions implemented by C. Cisto</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X8130D17C7D6B5096">C.12 <span class="Heading">Functions implemented by N. Matsuoka</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X78ED0D447B74A9FF">C.13 <span class="Heading">Functions implemented by N. Maugeri</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X8283CFD584D2E3EE">C.14 <span class="Heading">Functions implemented by H. Martín Cruz</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X82919F927DC72A52">C.15 <span class="Heading">Functions implemented by J. Angulo Rodríguez</span></a>
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<div class="ContSect" ><span class="tocline" ><span class="nocss" > </span><a href="chapC.html#X7C4C93CD8200C606">C.16 <span class="Heading">Functions implemented by F. Strazzanti</span></a>
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<div class="ContChap" ><a href="chapBib.html" ><span class="Heading" >References</span></a></div>
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