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<div class="pcenter">

<h1><strong class="pkg">numericalsgps</strong>-- a package for numerical semigroups</h1>

>

/>
<p><b> Manuel Delgado
    
    
  </b>
<br />Email: <span class="URL"><a java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 4
br /Homepage <span =U"<a ="ttp:/wwwfc.up./cmupmdelgado"http:/wwwfc.ppt//<a<span>
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<br />Homepage: <span class="URL"><a href="http://www.ugr.es/~pedro">http://www.ugr.es/~java.lang.StringIndexOutOfBoundsException: Index 90 out of bounds for length 0
<p>pb JoséJoã Morais
 
  </b>
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>© 2005--2015 Centroh3><

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<h3>Acknowledgements</h3>
 Centro Servicios de Informá yRedesde Comunicaciones (),Universidadde  for providing computingtime Rafael  Arredondo installingthis theextrasoftware inalhambraugres Santiago Ferrer in to cluster./java.lang.StringIndexOutOfBoundsException: Index 345 out of bounds for length 345

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<p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>The first and second authors warmly thank María Burgos java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 0

<p><strong class="button">Funding</strong></p>

<p>The first author

p>   byprojectsMTM2004-  MTM2007-,the  deAndalucí  343 FEDER ounds<pjava.lang.StringIndexOutOfBoundsException: Index 143 out of bounds for length 143

<p>The third author acknowledges financial support of java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<p>The first author was (partiallyjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p>Both maintainers java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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<p>The   asabbatical ://2018.>

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<p><a id="X7982162280BC7A61" name="X7982162280BC7A61"></pThisworkstartedwhen( )theauthor visitedthe of Granada inpartofa year Version . releasedin ) the maintained    authors  reportssuggestions are  coursewelcome useouremail addresses   effect<pjava.lang.StringIndexOutOfBoundsException: Index 321 out of bounds for length 321

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<p>If you have predominantly used the functions in the Appendix

<ph3Contentsaid="" ="">/>/>

<div class="contents">
<h3>java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">      Introduction
      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>
spanclass"> >spannocss>nbsp&;</span><a href=chap2html#X86D9D2EE7E1C16C2">2.1-2 NumericalSemigroupBySubAdditiveFunction</a></span>
<span""<r/<class=nocss"&;nbsp<spanahref=".#X799AC8727DB61A99">2.1-3 NumericalSemigroupByAperyList</a></span>
span =ContSS>< /< class=>nbsp&;span< hrefchap2html#X81A7E3527998A74A">2.1-4 NumericalSemigroupBySmallElements</a></span>
<panclass;<span< "html
<span class="ContSS"><br /><span class="<span class="ContSS"><br /><span class="nocss;&;/>< =chap2htmlX86AC8B0E7C11147F">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"><<span ="ContSS"<br >sjava.lang.StringIndexOutOfBoundsException: Index 147 out of bounds for length 147
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<span class="ContSS"><br /><span class="nocss"<class""<r/<spanclass"">nbsp;nbsp<span<href=chap2#X87B02A9F7AF90CB9">2.2-2 RepresentsSmallElementsOfNumericalSemigroup</a></span>
/></java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
/>
<div class="ContChap"><a href="chap3.html#X7A9D13C778697F6C">3 <span class="Heading">
                Basic operations withspanclass="ContSS>br/<pan=nocss"&;nbsp<spanahref=".
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< class=ContSect"< class"ocline>spanclass"">nbsp<span>a ="hap3htmlX87AF9D4F7FD9E820">3.1 <span class="Heading">
                    Invariants
                </span></a>
</</div>
<div class="ContSSBlock">
<spanclass=ContChap"ahref=chap3html#X7A9D13C778697F6C">3 <span class="Heading">
<span =""< >span class=nocss>nbsp;<span< "htmlX850F430A8284DF9A>.1- <a<span>
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span "ContSS>br/<class=nocss"&;nbsp<span< =chap3html#X84A6B16E8113167B">3.1-4 SmallElements</a></span>
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spanclassContSS"<r/span ">nbsp;nbsp<span<ahref=chap3html#X80431F487C71D67B">3.1-16 AperyList</a></span>
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<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>
<spanclass=ContSS"</>span class=">nbsp;/>a=chap3#X865E2E12804CFCD33125</>/>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X803D550C78717A7C">3.1-35 SpecialGaps</a></span>
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<div class="ContSSBlock">
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X865EA8377D632F53AddSpecialGapOfNumericalSemigroup/>/>
</div></div>
<div>
<div class="ContChap"><a href="chap4.html#X7969F7F27AAF0BF1">4 <span class="Heading">
               of  
            </span></a>
<div class="div class="ContSSBlock
n>
<div class="<span class="ontSS> /<spanclass">nbsp;nbsp<span><a =chap5html#X7F308BCE7A0E9D91"<>52- /<a/pan
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X815C0AF17A371E3E">4.1-3 BettiElements</a></span>
<pan =C"< /<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- AsNumericalDuplication</a></span>
<spanclass="</<span="nocss>nbsp;nbsp/a=chap4html>4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</java.lang.StringIndexOutOfBoundsException: Index 168 out of bounds for length 168
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</iv>/iv
<div classclass=""<br /<class"&; </< href=chap3#>322<a<span>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> <class"ontSS" /< class="ocss>&;;<span< href=".XF90E7EA43C47.24EliahouSlicesOfNumericalSemigroupa<span>
</</div
<< ""><ahref=chap4.X7969F7F27AAF0BF1">4 <span class="Heading">
</span>
<div class="ContSSBlock                Presentations of Numerical Semigroups
<span class="ContSS"><br /><span class="java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 23
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79C010537C838154">4.3-2 IsGeneric</a></span>
<java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
</div>
<div class="ContChap"><a href="chap5.html#X8148F05A830EE2D5">5 <span class="Heading">
                Constructing semigroups 
            </span></a>
<ivclass=ContSect>span "tocline"< nocss"&;/< href".#">51<java.lang.StringIndexOutOfBoundsException: Range [126, 125) out of bounds for length 142
                    <span class="ContSS"><br>span =">nbsp; <span>ahref=chap4.htmlX8750A6837EF75CA2"415<a<java.lang.StringIndexOutOfBoundsException: Range [161, 160) out of bounds for length 161
                </span></a>
</span>
<div class="ContSSBlock">
<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>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7DC65D547FB274D8">5.2 <span class="Heading">Intersections, sums, quotients, dilatations, numerical duplications and multiples by integers</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X875A8D2679153D4B">5.2-1 Intersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7F308BCE7A0E9D91"><code>5.2-2 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83CCE63C82F34C25">5.2-3 QuotientOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BE8DD6884DE693F">5.2-4 MultipleOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7F395079839BBE9D">5.2-5 NumericalDuplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8176CEB4829084B4">5.2-6 AsNumericalDuplication</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DCEC67A82130CD8">5.2-7 InductiveNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X81632C597E3E3DFE">5.2-8 DilatationOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X867D9A9A87CEB869">5.3 <span class="Heading">
                    Constructing the set of all numerical semigroups containing a given numerical semigroup
                
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><div "ContSect>span ="ocline"< ">n;/>ahref".#">.< classH>,java.lang.StringIndexOutOfBoundsException: Range [162, 161) out of bounds for length 246
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<span class="ContSS"><br /><span class="nocss">  </span>spanclass=""><br /span="ocss"nbsp; <>< =".html#X7BE8DD6884DE693F">.- MultipleOfNumericalSemigroup/><span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87369D567AA6DBA0">5.4-3 NumericalSemigroupsWithFrobeniusNumber</a></span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7DA1FA7780684019">5.5-3 NumericalSemigroupsWithMaxPrimitivePCspan>
/<div>
<div class="ContSect"><span class="tocline"><span class="<span class="Heading">
                    Constructing the set of numerical semigroups with genus g
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86970F6A868DEA95">5.6-1 NumericalSemigroupsWithGenus</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B4F3B5E841E3853">5.6-2 NumericalSemigroupsWithGenusPC</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8265233586477CC7">5.7 <span class="Heading">
                    Constructing the set of numerical semigroups with a given set of pseudo-Frobenius numbers
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X874B252180BD7EB4">5.7-1 ForcedIntegersForPseudoFrobenius</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87AAFFF9814E9BD2">5.7-2 SimpleForcedIntegersForPseudoFrobenius</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7D6775A57B800892">5.7-3 NumericalSemigroupsWithPseudoFrobeniusNumbers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X862DBFA379D52E2C">5.7-4 ANumericalSemigroupWithPseudoFrobeniusNumbers</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6.html#X83C597EC7FAA1C0F">6 <span class="Heading">
                Irreducible numerical semigroups
            </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X83C597EC7FAA1C0F">6.1 <span class="Heading">
                    Irreducible numerical semigroups
                </span></a>
</span>
<div class="ContSSBlock">
<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>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7EA0D85085C4B607">6.1-3 IsPseudoSymmetric</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#X87C2738C7AA109DC">6.1-6 IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8753F78D7FD732E2">6.1-7 DecomposeIntoIrreducibles</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7D3FD9C8786B5D72">6.2 <span class="Heading">
                    Complete intersection numerical semigroups
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X848FCB49851D19B8">6.2-1 AsGluingOfNumericalSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A0DF10F85F32194">6.2-2 IsCompleteIntersection</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86350BCE7D047599">6.2-3 CompleteIntersectionNumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7CD2A77778432E7B">6.2-4 IsFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86B4BA6A79F734A8">6.2-5 FreeNumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X830D0E0F7B8C6284">6.2-6 IsTelescopic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X84475353846384E8">6.2-7 TelescopicNumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A1C2C737BC1C4CE">6.2-8 IsUniversallyFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X847CD0EF8452F771">6.2-9 IsNumericalSemigroupAssociatedIrreduciblePlanarCurveSingularity</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8784D11578C912F2">6.2-10 NumericalSemigroupsPlanarSingularityWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </spanX"5.4N</>/spanjava.lang.StringIndexOutOfBoundsException: Index 175 out of bounds for length 175
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86F52FB67F76D2CB">6.2-13 IsAperySetAlphaRectangular</a></span>
</div></ class=ContSS"< >span class- NumericalSemigroupsWithFrobeniusNumberPC>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7998FF857F70C9A2">6.3 <span class="Heading">
                    numerical
                </span></a>
</span>
<divclass"ontSSBlock>
<span class="ContSS"><br /><span class="nocss">  </span><a href<span class=ContSS"br/< class""&&bsp<span>ahref=chap5htmlX7C17AB04877559B6>5.- NumericalSemigroupsWithMaxPrimitiveAndMultiplicity/a><span>
span "><br<span class="nocss>nbsp;n;<>< href".X8788F6597DBC6D98>6.2AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType/a<span
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< Constructing set   genusjava.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
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                    Several generalizing  conceptsymmetry
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8221EC44802E5158">6.4-1 IsGeneralizedGorenstein</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X866E48B47D66CFF2">6.4-2 IsNearlyGorenstein</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78049FC380A0006E">6.4-3 NearlyGorensteinVectors</a></span>
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</div></div>
</div>
<div class="ContChap"><a href="chap7.html#X83C2F0CF825B3869">7 <span class="Heading">
                Ideals of <divclass=ContChap"><a href="chap6.html#X83C597EC7FAA1C0F">6 <span class="Heading">
            </span></a>
<div class=ContSect>span classtocline"<span class="nocss"> </span><a href="chap7.html#X84B6453A8015B40B">7.1 <span class="Heading">
                     and
                </span></a>
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<ivclass>
<span class="ContSS"><br /><span class="nocss">  </span>< Constructingset all agivennumerical
<span class="ContSS"><br /<span
<span class="ContSS"><<pan classContSS>br>span class="nocss">  </span><a href="chap5.html#X7FBA34637ADAFEDA">5.3-1 OverSemigroups</a></span>
span ">br /<spanclass=">&&;<span< ref".X7A842A4385B73C63">.-Generators<a</pan>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81E445518529C175">7.1-5 AmbientNumericalSemigroupOfIdeal</a></span>
<class""><br/< class=>n&;span>a="chap7.">.16IsIntegral>/spanjava.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 132
span class"ontSS>< /<span "" &;<span>chap7.java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 149
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87369D567AA6DBA0NumericalSemigroupsWithFrobeniusNumber>/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7811E92487110941">7.1-9 SmallElements</a></span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87C2738C7AA109DC">6.1-6 IrreducibleNumericalSemi</a></span>
<span class="ContSS"><br /><span class="nocss">  </</div><div
< class"ontSS"< >span nocss"> nbsp</span<a href"chap7html#"7.1-11 FrobeniusNumber</a></span>
bsp; <spana "h#"7112PseudoFrobenius<a</java.lang.StringIndexOutOfBoundsException: Index 138 out of bounds for length 138
<span class=</span/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X821919B47D3D191A/span>
<span class="ContSS"><br /><span class="nocss<spanclass="ContSS">< > class=nocss>nbsp&;<span>ahref"htmlX83E8CC8F862D1FC0.- </>/>
< ">< /< ">&&java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 162
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B8B46CF7E81513D">7.1-17 NumberElement_IdealOfNumericalSemigroup</a></span>
< class"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 167
<></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A55BD4D82580537">7.1-20 Iterator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href< ="ontSS>br >< lass=nocss"&&/>a"htmlX7C8AB03F7E0B71F0"1- <a<span
< class"ContSS"><br/< class=nocss>nbspnbsp;/>ahref".html#X857FE5C57EE98F5E">7.1-22 MultipleOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X78743CE2845B5860">7.1-23 SubtractIdealsOfNumericalSemigroup</a></span>
< class=ContSS"<br><span "nocss>bsp;nbsp;/spana ".html#"71-4</><span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X803921F97BEDCA88">7.1-25 TranslationOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7CD66453842CD0AD">7.1-26 Union</a></span>
<">7.1-27 Intersection</a</>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D77F1BA7F22DA70">7.1-28 MaximalIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85975C3C86C2BC53">7.1-29 CanonicalIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D15FA4C843A13B7">7.1-30 IsCanonicalIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X829C9685798BB553">7.1-31 IsAlmostCanonicalIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X811B096B87636B8E">7.1-32 TraceIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7BB2A1B28139AA7E">7.1-33 TypeSequence</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X7F09B9A085E226EF">7.2 <span class="Heading">
        Decomposition into irreducibles
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B83DEAC866B65E8">7.2-1 IrreducibleZComponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83E064C684FA534C">7.2-2 DecomposeIntegralIdealIntoIrreducibles classC"/>span =nocss">;nbsp;/spana href".X86F52FB67F76D2CB>.13IsAperySetAlphaRectangular<span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X81CD9B12807EEA85">7.Almostsymmetric numerical 
                    Blow 
                </span></a>
</span>
<div class="ContSSBlock">
<=">br>spanclass""&;&</>a =chap7X82156F18807B00BF>.31</>span>
<span class="ContSS"><br /><span class="nocss">  </span><a hrefspanclass""< >spanclass"" &<span<href=chap6.#X84C44C7A7D9270BB>- <//>
< ="ontSS></<class=nocss" &</pana="html">.3- lowUp<a<spanjava.lang.StringIndexOutOfBoundsException: Index 128 out of bounds for length 128
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7FAABCBF8299B12F">7.3-4 ReductionNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a                    Several approaches generalizingthe concept symmetry
<span /java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<span=">br/<class="> ahref=chap7
< =""<br>span"">&;&;/span>< hrefchap7#"7- </>/
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7<span ""< /<class=nocss>nbsp;nbsp;/< href"chap7html#X87508E7A7CFB0B20>.1 BelongsToIdealOfNumericalSemigroup/
<</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X87AC917578976B1E">7.3-11 MicroInvariants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X805C984685EBC65C">7.3-12 AperyList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X82D2784B813C67D8">7.3-13 AperyList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8244CCAE7D957F46">7.3-14 AperyTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A16238D7EDB2AB3">7.3-15 StarClosureOfIdealOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X78F124CC82E7B585">7.4 <span class="Heading">
    Patterns for ideals
</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X865042FD7EBD15EE">7.4-1 IsAdmissiblePattern</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7ED8306681407D0F">7.4-2 IsStronglyAdmissiblePattern</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X799542C57E4E0D5E">7.4-3 AsIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F13F7CB7FBCF006">7.4-4 BoundForConductorOfImageOfPattern</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F4E597278AF31C8">7.4-5 ApplyPatternToIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7CFDFF6D7B9B595B">7.4-6 ApplyPatternToNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7F9232047F85C4D8">7.4-7 IsAdmittedPatternByIdeal</a></span>
<span <span class"ontSS/><span =n"  span><a href="chap6.html#X866E48B47D66CFF2">6.4-2 IsNearlyGorenstein</a></span>
<>/>
< class""<class"ocline>span=nocss"nbsp;/span< href=chap7.X79C6CE8D7EF1632D>.5< class=""Graded associated ring  numericalsemigroupspan>/>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> &java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 6
<span class="ContSS"><br /><span class< class="ContSect"><span class="tocline">span class="ocss">nbsp;/span><ahref="hap7htmlX84B6453A8015B40B">7. spanclass="eading"java.lang.StringIndexOutOfBoundsException: Index 142 out of bounds for length 142
<span class="ContSS"><br /><< ="ContSSBlock"java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
< classContSS"</< class=n"nbsp&bsp;/< ".X7"754</>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A5752C0836370FA">7.5-5 IsGradedAssociatedRingNumericalSemigroupGorenstein</a></span>
<spanupCI</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap8.html#X7D2E70FC82D979D3">8 <span class="Heading">
Numerical maximaljava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
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<divclass""< ""<class="nocsshref"html#X7D2E70FC82D979D38. span=Heading>
                    Numerical semigroups with maximal embedding dimension
                </span></a>
</span<pan="> >span">nbsp&bsp;/span< href"chap7.tml#>.- IsIntegral/<span
div "">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8774724085D3371DIdealByDivisorClosedSet/</span
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 132
< class"ontSS> /< class"nocss"nbsp /< href=chap8.tml#X848FD3FA7DB2DD4C">.- </a<span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X82E40EFD83A4A186">8.2 <span class="Heading">
                    Numerical semigroups with the Arf property and Arf closures
                <span<ajava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="<span class="ContSS"<br /< class=="" nbsp;/>a href=chap7.#805149CA847F6461">7.- PseudoFrobenius/>/
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<span class="spanclass="ContSS"</><span ="ocss"&bsp&</<href=chap7.html#java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 130
<span>ahrefchap8htmlXCD144384FD55F3.2- <a<span
<span class="ContSS"><br /><span class="nocss">  </spanspanclass="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83D0996D811A35C6">7.1-16 ElementNumber_IdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X80A13F7C81463AE5">8.2-6 <span class=ContSS"</<class=nocss>n;nbsp;/span>ahref="chap7.X79DFDCA27D3268FD"<>7.1- span></span[<>\/]/>/a>spanjava.lang.StringIndexOutOfBoundsException: Index 167 out of bounds for length 167
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X80EB35C17C83694D">8.2-7 ArfNumericalSemigroupsWithGenusUpTo</a></span>
< ">br/> class">nbsp&</>< hrefchap8#X7EE73B2F813F7E85828 ArfNumericalSemigroupsWithGenusAndFrobeniusNumber><span
<span class="ContSS"><br /><span class="nocss">  </span><a
<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 href="chap9.html#X7FDB54217B15148F">9.1 <span class="Heading">
        Factorizations in Numerical Semigroups
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> spanclass="ontSS"<br /<spanclass">nbsp&bsp;/>a".X78743CE2845B5860>. </
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X87C9E03C818AE1AA">9.1-3 FactorizationsElementListWRTNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X813D2A3A83916A36">9.1-4 RClassesOfSetOfFactorizations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7C5EED6D852C24DD">9.1-5 LShapes</a></span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X801DA4247A0BEBDA">9.1-8 DenumerantFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7D91A9377DAFAE35">9.1-9 DenumerantIdeal</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap9.html#X846FEE457D4EC03D">9</<div
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<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>
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        java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 37
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emigroups
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<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"><
        Numerical classContSect">span=">span&;/>< =h#"83 =Heading">
    </span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7B894ED27D38E4B5">9/span>
<span class="ContSS"><br /><span class="nocss"> <span =C"><br ><span class="nocss">  </span><a href="chap8.html#X81CCD9A88127E549">8.3-1 IsSaturated</a></span>
<span="ontSS> />class"ocss"&; /><a href=".html">9.3</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8630DEF77A350D76">9.8-4 IsSuperSymmetricNumericalSemigroup</a></span>
<>/>
</div>
<div class="ContChap"><a href="chap10.html#X7D2C77607815273E">10 <span class="Heading">
  Polynomials and java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 6
 </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">        Invariants based on 
  java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 7
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</span>
java.lang.StringIndexOutOfBoundsException: Range [68, 69) out of bounds for length 25
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X834D6B1A7C421B9F">10.2-2 IsDeltaSequence</a></<pan classContSS">br/<spanclass"nocss" nbsp</span =".html#X7A2B01BB87086283>9-Elasticity<>/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X824ABFD680A34495">10.2-3 DeltaSequencesWithFrobeniusNumber</a></span>
n=ContSS">br< class="nocss>nbsp;nbsp;/<href=".#87B819B886CA5A5C".2 </>/span
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8597259279D1E793">10.2-7 SemigroupOfValuesOfCurve_Global</a></span>
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<class="ontSect<span class=tocline"< class="nocss>nbsp;</>a "hX84C670E1826F8B92">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#/
<span class="ContSS"><br /><span class="nocss">  </span>span class=ContSS"<br /<span class="nocss" nbsp;</span><a href="chap9.html#X83075D7F837ACCB8">9.4-1 OmegaPrimality</a></span>
<panclass="ContSS>br/>span class="nocss">  </span><a href="chap11.html#X7D4D017A79AD98E2">11.1-19 HilbertBasisOfSystemOfHomogeneousEquations</a></span>
<class=">br/<span class="">& span>a href=c.html#825B1CD37B0407A6"111-20 HilbertBasisOfSystemOfHomogeneousInequalities><span
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<div Homogenization of  Semigroups
    Gluings of affine semigroups
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<span class="ContSS"><br />span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X85D03DBB7BA3B1FB">9.5-2 FactorizationsInHomogenizationOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap11.html#X86A1018D7CB7BA81">11.3 <span class="Heading">
    Presentations of affine semigroups
  </span></a>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7C1B355F83A55285">11.3-8 ShadedSetOfElementInAffineSemigroup</a></span>
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    Factorizations in affine semigroups
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<span class="ContSS"><br  umerical semigroups with Apéry sets having special factorization properties
<span<span</java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<="ContSS<br />span classnocss>&;nbsp</><a href".X819CDBAA84DB7E83".-4 />/java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
<c=ContSS>br /<=n"&;;<pan>a href==".html"1145DeltaSet/>/java.lang.StringIndexOutOfBoundsException: Index 132 out of bounds for length 132
class"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 138
<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<a chap11#>.-10TameDegree<a</>
<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>java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<div class="ContSect"><span class="tocline"><spanspanclass=ContSS"</<spanclass=nocss"&&;/span "html#>10.17 IsKroneckerPolynomial<</>
java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 52
  </span></ah#X7B428FA2877EC7331019 /a>
java.lang.StringIndexOutOfBoundsException: Range [58, 59) out of bounds for length 7
< class=ContSSBlock"
<span class="ContSS"><br /><span class="nocss">  /div<div>
<span class="ContSS""<span class""<spanclass">nbsp;/< href"htmlXE">
<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-java.lang.StringIndexOutOfBoundsException: Range [139, 137) out of bounds for length 139
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X79ECACE4793A6B00">11.5-13 MaximalIdeal</a></span>
</div></div>
</div>
< class"ontChapa =.html#"> span class">
  Good semigroups
java.lang.StringIndexOutOfBoundsException: Index 104 out of bounds for length 12
<div class="ContSect"><span=""<br/spanclass"nocss;nbsp;</>ahref="java.lang.StringIndexOutOfBoundsException: Range [82, 81) out of bounds for length 155

    </span></a>
<span>
<div class="ContSSBlock">
<ContSS>< ><span java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 139
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X82A8863E78650FC4">12.1-2 NumericalSemigroupDuplication</a></span>
<span classdivclass="ContSect>span ="><span ="&;/span<href"chap10h#>103 span =">
<span class="ContSS"><br /><span class="nocss">  </spanspan</>
<spandivclass=C"
</iv</iv>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X8431465B82643392">12.2 <span class="Heading">
      Notable elements
  />/java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
</span>
<div class="ContSSBlock">
<bsp;</spanahref=chap12.html#>.2-1 </a<spanjava.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 146
<span class="ContSSspan></java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<span class=<iv="ContSSBlock">
<spanclassContSS<br/< "ocss">nbsp;nbsp;<<a ="hap12.tml#"122- /><span>
<span class="ContSS"><br /><span class="<spanclass="ontSS"< ><java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 150
<span class="ContSS"><br /><span class="nocss">  </span><a span=ContSS"</span =nocss">;nbsp<span>a=chap11.html#X7846AD1081C14EF111. <a<>
7 GoodSemigroupBySmallElementsa>/pan>
<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>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap12.html#X87FE42227F47666F">12.3 <span class="Heading">
      Symmetric 
  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><spanspan class""<br ><span class="">nbsp;nbsp;/><ahref=chap11.#X7A1CE5A98425CEA111.-22BasisOfGroupGivenByEquations</>
</div></div>
<="tocline">class="&<span>a chap12.#X80A3D64386A152EB>.4<spanclass=H"java.lang.StringIndexOutOfBoundsException: Index 144 out of bounds for length 144
    Arf good closure
  </span></a>
<spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<div class==ContSSBlock">
<span "ontSS>br /span "">nbsp;/>a ="hap12.tml#">.41 ArfClosure<a</pan>
</div></div>
<div     resentations  
      Good ideals
    </span></
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X843CA9D5874A33F2">12.5-1 GoodIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#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="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X84636A127ECEDA24">12.5-4 MinimalGoodGeneratingSystemOfGoodIdealspan class=""><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="chap12.html#X797999937E4E1E2B">12.5-5 BelongsToGoodIdeal</a></span>
< classclass="ontSS>br /><span =n"> &;</span<a "chap12.#X842F3CE07E893949>.5-6 SmallElements</a</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7DA7AE32837CC1C7">12.5-7 CanonicalIdealOfGoodSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X7DC7A4B57BC2E55C">12.5-8 AbsoluteIrreduciblesOfGoodSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X87AB3B09857B383A">12.5-9 TracksOfGoodSemigroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap13.html#X84A2793F7A9F3E6A">13 <span class="Heading">
  External packages
 </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap13.html#X7BD18FC581F0C4D3">13.1 <span class="Heading">
          packages
    </span></a>
</span>
<class=ContSSBlock"
<span class="ContSS"><br /><span class="nocssspan="ContSS"<br>spanclass=""> &;/span>< href="hap11#X8457595E7AA542E6"114 TameDegree/<span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X875001717A8CF032">13.1-2 NumSgpsUse4ti2gap</a></span>
<><a href"h#>13.13 <a>/>
<span class="ContSS"><br /><span class="nocss">  </span><a <span class="ContSS"><br /><span class="nocss">  </span><a hrefhtmlX7A3571E187D0FCDE">14- OmegaPrimalitya</span
<="ontSS>/><spanclass=nocss>nbsp;nbsp;/>a href=chap13.html#">13-5 NumSgpsUseSingularInterface/>/>
</div></div>
</div>
<div   <>/java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
  Dot functions
 <<ajava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap14.html#X7B8D661F79E957A6">14.1 <span class="Heading">
      Dot functions
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7FEF6EC77E489886">14.1-1 DotBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X868991B084E42CE9">14.1-2 HasseDiagramOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X832901FF85EB8F1C">14.1-3 HasseDiagramOfBettiElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8050862F79EA9620">14.1-4 HasseDiagramOfAperyListOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7F62870F8652EDE6">14.1-5 DotTreeOfGluingsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7F43955582F472B6">14.1-6 DotOverSemigroupsNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X8195A2027B726448">14.1-7 DotRosalesGraph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7EC75F477D4F8CC3">14.1-8 DotFactorizationGraph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X83F1423980D2AEA4">14.1-9 DotEliahouGraph</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X81F579B783CF4363">14.1-10 SetDotNSEngine</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap14.html#X7D1999A88268979F">14.1-11 DotSplash</a></span>
</div></div>
</div>
<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>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X86859C84858ECAF1">A.1-1 BezoutSequence</a></span>
spanclass"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"< class"ontSS>br >< class"nocss">  </span><a href="chap11.html#X7F16A5A27CBB7B93">11.5-3 MinimalGenerators</a></span>
<></>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7D3D347987953F44">A.2 <span class="Heading">Periodic subadditivebr /><span class="nocss">  </span><a href="chap11.html#X81CA53DA8216DC82">11.3-9 IsGeneric</a></span>
</span>
<div class="ContSSBlock">
<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>
</div></div>
</div>
<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>
</div></div>
<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>
</div></div>
<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>
</div></div>
</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>
</div>
<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>
/
</div>
<div  <span></>
<<span>
</div>
<div class< class">br/<class"">;nbsp</>a =c.
</span>
</div>
div "< =tocline"< "nocss"&;/>ahref".#X7FAE71B27B0E3889">C.7 <span class="Heading">Functions implemented by G. Zito</span></a>
</span>
</div>
<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>
<<spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
</div>
<div class="ContSect"><< class"">span ="< class=nocss"&;<span<a =chap12.#X7FA8DCAC7951F7FB">12.5 <span class="Heading">
</span>
</div>
<div class="ContSect"><span class="tocline"><span <iv class=ContSSBlock"
</span>
</div>
<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>
</span>
</div>
<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>
</span>
</div>
<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>
</span>
</div>
<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>
</span>
</div>
<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>
</span>
</div>
<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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