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<h1><strong class="pkg">numericalsgps</strong>-- a package java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 7

<p> Version 1.4.0</p>

</div>
<p><b> Manuel Delgado
    < class=chlinkprevnexttop>&bsp<href=chap0.">Top  Book]<a>nbsp;<ahref=".#">Contents]</a>    <a href="chap1.html">[Next Chapter]</a>   </div>
    
  </b>
<br />Email: <span class="URL"><a href="mailto:mdelgado@fc.<>a" java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 62
<br />Homepage: <<p> Version 1.4.0</p
<<divjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
    
    
  </b>
<br />Email: <span class="URL"><a href="mailto:pedro@ugr.es">pedro<>:class"RL>ahref="/.fcuppt/>/www.fcu.cmupmdelgado</>/
<span =URL>ahref"ttp:/wwwugr.es/pedro">:/wwugr./pedro</<span>
</p><p><b> José João Morais
 
  </b>
</p>

<p><a id="X81488B807F2A1CF1
<3Copyright/>
<p>© 2005--2015 Centro de Matemática da Universidade<br/Email span class">a href":@ugr.es>@ugr.s<a><sjava.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 84

<p><em>Numericalsgps</em> is free software; you /<><>  Joojava.lang.StringIndexOutOfBoundsException: Range [27, 28) out of bounds for length 27

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<Acknowledgements<h3>
<p>

<p>The maintainers want to thank the java.lang.StringIndexOutOfBoundsException: Range [37, 0) out of bounds for length 0

<p>The authors also thankthe  deServiciosticay  de  CSIRC  Granada,for thecomputing ,specially Arco forinstalling  packageand    needed .ugr.,and SantiagoMelchor  forhelping  jobsubmission  the<p>

<p>The first and second

<p><strong class="button">Funding<

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

<p>The second author was supported by the projects MTM2004-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

<Thesecond authorwassupported  the  01446andMTM200762346 theJunta agroupFQM-,and founds./>

<p>Both maintainers were (java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

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

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

<p>The first author acknowledges a sabbatical grant from the FCT: SFRH/

<p>The second author was supported in part by Grant PGC2018

Thefirstauthoracknowledges a sabbatical grant from theFCT SFRHBSAB142918/</pjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92

<p>Both maintainers acknowledge the "Proyecto de Excelencia de la java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 0

<p><a id="X7982162280BC7A61" 
h3Colophon<h3java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 17
<>    in2004  first  theUniversity     sabbatical.Since 0.6( in2008, packageis  bythefirsttwo.Bug ,  andcomments ,of, .Please  our  addressestothiseffect./>

<p>If you have benefited from the use of the

<p>If you have predominantly used the 

<p><a id="X8537FEB07AF2BEC8" name=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div class="contents"
<>< =contentsname"ontents><a<h3java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 54

<div class="ContChap"><a href=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
      
    </span></a>
java.lang.StringIndexOutOfBoundsException: Index 140 out of bounds for length 140
<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< class=ContSS"<br/< class=""&;nbsphref".html
<span< class=ContSS>< >span class">nbsp&;/>< chap2html
<span class="ContSS"><class="br >span"nocss"&;nbsp</>ahref=".
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X879171CD7AC80BB5">2.1-9 ProportionallyModularNumericalSemigroup</a></span>
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<span class="ContSS"><br /><span < ""</<classnocss"nbsp&;/< href"chap3htmlX81A2505E8120F4D7"><code>3.1-8 <span>\</span>[ <span>\</span>]</code></a></span>
<span =ContSS>br/<panclass""&;nbsp</span<href".#X850F430A8284DF9A">3.1-2 Generators</a></span>
">br /< ""&;nbsp<span>ahref=chap3.html#X7884AE27790E687F">3.1-3 EmbeddingDimension</a></span>
spanclass"ContSS>br/< class=">n;n;/>ahref".htmljava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
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span class=ContSS"< >span=nocss>nbsp;nbsp<span>ahref".#java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
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< ""br/< class">&<spanahref=chap3htmlX7B6C82DD86E5422F">3.1-11 ElementNumber_NumericalSemigroup</a></span>
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<span class="ContSS"><br /><span class="nocss">  </span><a =">br >classnocss"&; <span< href".html#">.- Type<a</span
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<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">  <<pan="< >span =nocss>nbsp&;<span>ahref"htmlX7E9C8E157C4EAAB0"3- /><span>
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<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 "ContSS>br > class"&&;<span>< href".#X7A9B5AE782CAEA2F"</a>span>
</span>
<div""
<span class="</iv></>
<span class"ontSS>< />span ="nocss>nbspnbsp;span>a".htmlX80F9EC9A7BF4E606".- EliahouNumber/>/
<span java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
span class=""<br>span ""&bsp />a href"chap3html#7846">3- </>/>
</div></div>
>
<divclass=ContChap< href"chap4.html#java.lang.StringIndexOutOfBoundsException: Range [60, 59) out of bounds for length 85
                 java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 53
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<div class="ContSect"><span class="tocline"><span class="spanjava.lang.StringIndexOutOfBoundsException: Range [19, 11) out of bounds for length 131
</span>
<div class="ContSSBlock">
<span /div>
<span class="ContSS"><br /><span class="nocss">  </java.lang.StringIndexOutOfBoundsException: Range [0, 65) out of bounds for length 6
<span class="ContSS"><br /><                numerical  fromothers
<span class="ContSS"><br /><d ""< class=">spanclass="nocss>nbsp<span>ahref=chap5.tmlX782F3AB97ACF84B85. span class="Heading">
< /<class"ocss" nbsp;/< "#>.- AllMinimalRelationsOfNumericalSemigroup/>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="java.lang.StringIndexOutOfBoundsException: Range [16, 80) out of bounds for length 27
<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="nocss"> </span><a href="chap5.html#X782F3AB97ACF84B8">5.1 <span class="Heading">
                    Adding and removing elements of a numerical semigroup
                </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>
<class="ContSect"< classtocline>spanclass=nocss"&bsp<span< href=chap5htmlX7DC65D547FB274D852 span="eading"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./div>/>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X83CCE63C82F34C25">5.2-3 QuotientOfNumericalSemigroup</a></span>
span ContSS >< class=n> &;/spanahrefchap5html"524</>/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7F395079839BBE9D"span =ContSS"< >< class"ocss"nbsp&> =chap5.X7DB3994B872C4940>.4 NumericalSemigroupsWithFrobeniusNumberAndMultiplicity</a></java.lang.StringIndexOutOfBoundsException: Index 175 out of bounds for length 175
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8176CEB4829084B4">5. NumericalSemigroupsWithFrobeniusNumberPC><span>
<span d ="ContSect"spanclass="<class=n>nbsp;</span><a href="chap5.html#X8021419483185FE3">5.5 <span class="Heading"> Constructing the set of numerical semigroups with given maximum primitive</span></</>
< classContSS</>span"nocss">  </span><a href="chap5.html#X81632C597E3E3DFE">5.2-8 DilatationOfNumericalSemigroup</a></span>
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<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></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">&bsp&;/>a href=chap5htmlX7FBA34637ADAFEDA>3- OverSemigroups<a<span>
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<div class="ContSect"><span class="tocline"><span class="nocss"><span class=ContSS>br/<span class=n"  /<a href="chap5.html#X875A8B337DFA01F0">5.5-2 NumericalSemigroupsWithMaxPrimitive</a></span>
</>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  <div>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#7DB3994B872C4940>54-2 umericalSemigroupsWithFrobeniusNumberAndMultiplicitya<span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X87369D567AA6DBA0">5.4<span class=ContSS>br /><span class="nocss">  </span><a href="chap6.html#X7E6E262C7C421635">6.2-12 IsAperySetBetaRectangular</a></span>
<span"ContSS>br/<class="nocss">  </span><a href="chap5.html#X80CACB287B4609E1">5.44</a></spanjava.lang.StringIndexOutOfBoundsException: Index 162 out of bounds for length 162
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a Almost-symmetric numerical semigroups
</span>
<div class="<div class="ContSSBlock =C"java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds for length 25
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<span class="ContSS"><br /><span class="nocss">  </span><a href=<class=ContSS" />class=nocss"&&bsp/spana=chap6html#"3- AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType<a>/>
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</div></div>
<div="ContSect>tocline"< "nocss"&;/><ahref"hap5.X7D6635CB7D041A54"56< ""
                     the ofnumerical semigroupswith g
                </span></a>
</span>
<div div class>span =tocline> class"ocss>nbsp<spana href=html#X7FDC79A285EE016B">6.4 <span class="Heading">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86970F6A868DEA95">5.6-1                      approachesgeneralizingthe of 
<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"> &java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 12
</div></div>
</div>
<"java.lang.StringIndexOutOfBoundsException: Range [21, 20) out of bounds for length 85
                Irreducible numerical semigroups
            </span></a>
<div class="ContSect"><span class="tocline"><<span class="ContSS"><br /><span class="nocss">&n;/><ahref=chap7.tmlX805149CA847F6461>.- /a>span>
                    Irreducible numerical semigroups
                ><ajava.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
<java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<div class="ContSSBlock">
span class"ContSS">r/<span""&;nbsp;/< =chap6.#">611IsIrreduciblea<span
<span class="ContSS"><br /><span class="nocss">  </span><aspanclass=ContSS"br/spanclass=nocss" nbsp;</span><a href="chap7.html#X83D0996D811A35C6">7.1-16 ElementNumber_IdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html<spanclass=ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79DFDCA27D3268FD"><code>7.1-18 <span>\</span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8341AE847D005E9F"><code>7.1-19 \{ \}</code
<spanclass""</spanc">nbsp;nbsp;<span< href=chap6.#X7C8AB03F7E0B71F0">.-AnIrreducibleNumericalSemigroupWithFrobeniusNumber/a>/>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X78345A267ADEFBAB">6.spanclass= >span""&;&<span>a =chap7java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 158
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X87C2738C7AA109DC">6.1-6 IrreducibleNumericalSemigroupsWithFrobeniusNumberAndMultiplicity</<pan"> /<spanclass=">nbsp&bsp<span><href=chap7X8321A10885D2DEF8>.- Differencea<
<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.span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B34033979009F64>/spanjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
                    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">  </span><a href="chap6.html#X80CAA1FA7F6FF4FD">6.2-11 IsAperySetGammaRectangular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7E6E262C7C421635">6.2-12 IsAperySetBetaRectangular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X86F52FB67F76D2CB">6.2-13 IsAperySetAlphaRectangular</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7998FF857F70C9A2">6.3 <span class="Heading">
                    Almost-symmetric numerical semigroups
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7A81F31479DB5DF2">6.3-1 AlmostSymmetricNumericalSemigroupsFromIrreducible</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X8788F6597DBC6D98">6.3-2 AlmostSymmetricNumericalSemigroupsFromIrreducibleAndGivenType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X84C44C7A7D9270BB">6.3-3 IsAlmostSymmetric</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X7B0DF2FE7D00A9E0">6.3-4 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X85C4DA6E82E726D2">6.3-5 AlmostSymmetricNumericalSemigroupsWithFrobeniusNumberAndType</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6.html#X7FDC79A285EE016B">6.4 <span class="Heading">
                    Several approaches generalizing the concept of symmetry
                </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>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6.html#X83F13D6482F021B2">6.4-4 IsGeneralizedAlmostSymmetric</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap7.html#X83C2F0CF825B3869">7 <span class="Heading">
                Ideals of numerical semigroups
            </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X84B6453A8015B40B">7.1 <span class="Heading">
                    Definitions and basic operations
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X78E5F44E81485C17">7.1-1 IdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85BD6FAD7EA3B5DD">7.1-2 IsIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X85144E0F791038AE">7.1-3 MinimalGenerators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A842A4385B73C63">7.1-4 Generators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81E445518529C175">7.1-5 AmbientNumericalSemigroupOfIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B0343BF794AC7EA">7.1-6 IsIntegral</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X80233A6F80CA0615">7.1-7 IsComplementOfIntegralIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8774724085D3371D">7.1-8 IdealByDivisorClosedSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7811E92487110941">7.1-9 SmallElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7EDDC78883A98A6E">7.1-10 Conductor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A8AF91C7D1F1B4E">7.1-11 FrobeniusNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X805149CA847F6461">7.1-12 PseudoFrobenius</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D4C7C997EEAADF7">7.1-13 Type</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a spanclassContSS>br /< class=nocss&bsp <>a=".htmlX82C2329380B9882D>.3-RatliffRushClosurea<span>
<spanclass=ContSS>br /span ""&&<span>a=chap7."7-15</a><span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X83D0996D811A35C6">7.1-16 span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8344B30D7EDE3B04">7.3-10 MultiplicitySequence
<span class="ContSS"><br /><span class="nocss">  </span>< href="chap8.html#X7CC73F15831B06CE">8.2-9 ArfSpecialGaps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79DFDCA27D3268FD"><code>7.1-18 <span>\</span>[ <span>\</span>]</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8341AE847D005E9F"><code>7.1-19 \{ \}</code></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="chap7.html#X7B39610D7AD5A654">7.1-21 SumIdealsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.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>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8321A10885D2DEF8">7.1-24 Difference</a></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>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7B34033979009F64">7.1-27 Intersection</a></span>
<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</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X81CD9B12807EEA85">7.3 <span class="Heading">
                    Blow ups and closures
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X82156F18807B00BF">7.3-1 HilbertFunctionOfIdealOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X81F1F3EB868D2117">7.3-2 HilbertFunction</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79A1A22D8615BF78">7.3-3 BlowUp</a></span>
<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 href="chap7.html#X7BFC52B7804542F5">7.3-5 BlowUp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8799F0347FF0D510">7.3-6 LipmanSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7D6F643687DF8724">7.3-7 RatliffRushNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X82C2329380B9882D">7.3-8 RatliffRushClosure</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X79494A587A549E15">7.3-9 AsymptoticRatliffRushNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X8344B30D7EDE3B04">7.3-10 MultiplicitySequence</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 class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X827BB22083390CB9">7.4-8 IsAdmittedPatternByNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap7.html#X79C6CE8D7EF1632D">7.5 <span class="Heading">Graded associated ring of numerical semigroup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7876199778D6B320">7.5-1 IsGradedAssociatedRingNumericalSemigroupCM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X782D557583CEDD04">7.5-2 IsGradedAssociatedRingNumericalSemigroupBuchsbaum</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X78E57B9982F6E1DC">7.5-3 TorsionOfAssociatedGradedRingNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7E16B6947BE375B2">7.5-4 BuchsbaumNumberOfAssociatedGradedRingNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7A5752C0836370FA">7.5-5 IsGradedAssociatedRingNumericalSemigroupGorenstein</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap7.html#X7800C5D68641E2B7">7.5-6 IsGradedAssociatedRingNumericalSemigroupCI</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap8.html#X7D2E70FC82D979D3">8 <span class="Heading">
                Numerical semigroups with maximal embedding dimension
            </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X7D2E70FC82D979D3">8.1 <span class="Heading">
                    Numerical semigroups with maximal embedding dimension
                </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X783A0BE786C6BBBE">8.1-1 IsMED</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7A6379A382D1FC20<pan classContSS> /><span class="nocss">  </span><a href="chap9.html#X7F4CBFF17BBB37DE">9.6-5 NumericalSemigroupByTauSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#java.lang.StringIndexOutOfBoundsException: Index 99 out of bounds for length 70
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap8.html#X82E40EFD83A4A186">8.2 <span class="Heading">
                     semigroups with the Arf property and Arf losures
                </span></a
<
<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0
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<span class="ContSS"><br />< ,  thatdistribution ispermitted only  oramong
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< =ContSS"b >span class="">&;nbsp;</<a href="chap8h#"82- rfNumericalSemigroupsWithGenus<a<sjava.lang.StringIndexOutOfBoundsException: Index 153 out of bounds for length 153
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< "ontSS"< /<span "nocss>nbsp;&;<span<href"chap8XEE73B2F813F7E85>82- <a><>
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<div =""< class"tocline"> class="nocss">nbsp<spanahref"chap8.tmlX7E6D857179E5BF1B>. <spanclass"java.lang.StringIndexOutOfBoundsException: Range [141, 140) out of bounds for length 142
                    Saturated numerical semigroups
                </span></a>
<span
<div class="ContSSBlock">
class"ontSS/<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 133
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X78E6F00287A23FC1">8. classC"<br>span =">nbsp<spana=chap9#X80B707EE79990E1E98- IsHomogeneousNumericalSemigroup
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X7CC07D997880E298">8.3-3 SaturatedNumericalSemigroupsWithFrobeniusNumber</a>/div><div>
</div></div>
</div>
<div class="ContChap"><a href="chap9.html#X7B6F914879CD505F">9 <span class="Heading">
  java.lang.StringIndexOutOfBoundsException: Range [20, 11) out of bounds for length 73
 </span><//
<="ontSect"span lasstocline> "ocss<s<="chap9.#7FDB54217B15148F. < class"">
java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 46
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</span>
<div class="span class="ContSS"><br /><span class="nocss">  </span><a href="chap8.html#X848FD3FA7DB2DD4C">8.1-3 MinimalMEDGeneratingSystemOfMEDNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X8429AECF78EE7EAB">9.1-1 FactorizationsIntegerWRTList</a></span>
<span class="ContSS/
<><br/<classnocss>nbsp&<<href=.tmlX87C9E03C818AE1AA913</>/
<span class="ContSS<spanclass="ContSS">br />< class=nocss"&bsp span>< =".X86137A2A7D27F7EC>2 IsArf<a<span
<="><br/>< =n">;nbsp;/<a href=chap9.tmlX7C5EED6D852C24DD">91- LShapes/>/span>
<span class="ContSS"><br /><span class="nocss"> &pan java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 158
<
<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">9.2 <span class="Heading">
        basedonlengths
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS<span<a>
<span class="ContSS"><br /><span class="nocss">  </span><a
<span class="ContSSspan "ontSS></<class="ocss>nbsp; </span><a href="chap10.html#X7FFF949A7BEEA912">10.2-1 SemigroupOfValuesOfPlaneCurveWithSinglePlaceAtInfinity</a></span>
< class=""<br /span class=>&&;/><ahref"chap9#".24 /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.">9.2-13 sAdditiveNumericalSemigroup</>/span>
<span class=ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X790308B07AB1A5C8">9.2-14 MaximalDenumerant</a></span>
span class"><br ><span class="nocss">  </span><a href="chap9.html#X7DFC4ED0827761C1">9.2-15 MaximalDenumerantOfSetOfFactorizations</a></span>
< class="ContSS">br/><spanclass=nocss" &bsp;/<href".tmlX811E5FFB83CCA4CE>9216 MaximalDenumerant</a</
<span class="ContSS"><br /><span class="nocss"> <div<div>
</divclass=">span class"> class"nocss&;< href"hap9.tml#java.lang.StringIndexOutOfBoundsException: Range [115, 114) out of bounds for length 142
<div class="ContSect"><span java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 15
        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>nbsp;java.lang.StringIndexOutOfBoundsException: Range [60, 58) out of bounds for length 136
<span class="ContSS"><br /><span class="nocss"> s classContSS"< /<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 167
<span class="ContSS"><br />span class"ContSS"<br >"nocssnbsp;nbsp</span>href"hap11htmlX">.- </a>/>
<div>/div>
<div class="ContSect"><span class="tocline"><span classspan class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7A1CE5A98425CEA1">11.1-22 BasisOfGroupGivenByEquations</a></span>
          NumericalSemigroups
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><div class="ContSSBlock
java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 172
<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 />
        Divisors, posets
    </span></aspan class="ContSS"<br /<span class="ocss>  <span><=chap11.tml#X795EEE4481E0497C>13- CircuitsOfKernelCongruencea<span>
<span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X853930E97F7F8A43">9.6-1 span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7EE005267DEBC1DE">11.3-3 GeneratorsOfKernelCongruence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#X7DF6825185C619AC">.62 MoebiusFunction<a<span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.html#span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7C3546477E07A1EA<>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap9.pan =ContSS"></span="">nbsp&;/span>ah=chap11html#X80A7BD7478D8A94A">11.3-6 MinimalPresentation</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><spanclass=""> >span =nocss"&&;</pan>a href=chap11.#X81CA53DA8216DC82">.- /a>/
java.lang.StringIndexOutOfBoundsException: Range [24, 8) out of bounds for length 38
    </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="div"ContSect><span "ocline<span =""nbsp;/pan href".htmlX80A934B0826E21A6>.4 spanclass=Heading"
<span class="ContSS"><br /><span />
</div></div>
< class="">span ="ocline< =nocss"&bsp<span>a =".html#X79A8A15087CEE8C1">9.8 span class"">
        java.lang.StringIndexOutOfBoundsException: Range [18, 17) out of bounds for length 84
    />a>
</span>
<div span class""><br>span="">nbsp&bsp;/pana=chap11html#X819CDBAA84DB7E83">1144 Elasticity<a<span>
<span classspan lass"ontSS"<br >span class"ocss>nbsp;nbsp;</pan> "hap11#X839549448300AD26".- DeltaSet</<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 java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 12
<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 href=.tmlX7D2C77607815273E10<java.lang.StringIndexOutOfBoundsException: Range [71, 70) out of bounds for length 87
  Polynomials and numerical class="<java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 142
 </span></a>
<div class="ContSect"><span/span><a href="chap9.html#X856B689185C1F5D9">9.5-1 BelongsToHomogenizationOfNumericalSemigroup</a></span>
  Generating functions or Hilbert series
 </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS<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="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 class="ContSS"><br /><span class="nocss"> >ahref="hap11.htmlX8457595E7AA542E6"114 <>/spanjava.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
<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>/>/>
< ">br >< =">nbsp;nbsp;<span><ahref=chap10.X7D9618ED83776B0B"10.1-7IsKroneckerPolynomial<a>/pan>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X8366BB727C496D31
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.tml#">.-9 CyclotomicExponentSequence<>/span>
<span class="ContSS"><br /><span class="nocss">  
<span class="ContSS"><br /><span class="nocss">  divclass">
<>/java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
">=tocline> class==nocss"&bsp<span>a=chap10.#7EF2A1781432A2D10.2 <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="div =C">< href=chap12htmlX7A9271AC84C7277F12<span class=Heading"
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X7F88774F7812D30E
<span class"ontSS> />< class=""> <span<a hrefchap10.html#X8597259279D1E793">10.2-7 SemigroupOfValuesOfCurve_Global</a></span>
<span class="ContSS"><br /><span 
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap10.html#X836D31F787641C22"/
<span class="ContSS"><br /><span class=span class=""br/<class="nocss">  </span><a href="chap12.html#X79E86DEE79281BF2">12.1-1 IsGoodSemigroup</a></span>
</div></div>
< "<class=tocline">class"nocss>nbsp;/span>a =.tmlX84C670E1826F8B92"10.3<spanclass==Heading
  Semigroups and Legendrian curves
 </span><a
</span>
< "ontSSBlock>
<span class="ContSS"></>/iv
</div></div>
</div>
<div class="ContChap"><a href="<span<a>
  Affine semigroups
 </span></a>
<div class="ContSect"><span class="tocline"><span span class="ContSS"><br /><span class="nocss"> &n&gt;< href=chap12.X79EBBF6D7A2C9A12"1221 BelongsToGoodSemigroupa>/>
        Defining affine semigroups
    </pana>
</span>
< classContSSBlock>
<span class="ContSSspan class=""> /spanclass=n"&&;<span> href="htmlX792BCCF87CF63122>.4 IsLocal<a/
< ">br/span class="nocss">  </span><a href="chap11.html#X855C8667830AEDDC">11.1-2 AffineSemigroupByEquations</a></span>
< class">br ><class=">nbsp&bsp;/>< href=".X7846AD1081C14EF1">111-3AffineSemigroupByInequalities/>/span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7CC110D4798AAD99">11.1-7GoodSemigroupBySmallElements</a</java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 152
<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">11java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<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>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7D4D017A79AD98E2">11.1-19 java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 12
<span class="ContSS"><br /><span class="nocss">  </span><a href goodsemigroups
<span class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 7
<=ContSS>br/<classnocss &<span< "chap11html#">122 </a>span
</div></div>
<div class="ContSect"><span <div class="ContSect"><span class>span ="nocss>nbsp;/<href="hap12html">12 span class"eading"
    Gluings of affine semigroups
  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss/>
<">
<div classclass=C"< ><class=nocss>nbsp;&<span><href"hX87248BD481228F36>12-1ArfClosure/a></java.lang.StringIndexOutOfBoundsException: Index 134 out of bounds for length 134
Pof affine semigroups
  </span>       
java.lang.StringIndexOutOfBoundsException: Range [13, 14) out of bounds for length 7
<div class="ContSSBlock">
<class"ontSSbr /<java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 150
<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 GeneratorsOfKernelCongruencespan "ontSS">>class"ocss nbsp;/span>ahref=".htmlX842F3CE07E893949"125-6  SmallElements>/span
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7AD2271E84F705D3">11.3-4 CanonicalBasisOfKernelCongruence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7C3546477E07A1EA">11.3-5 GraverBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X80A7BD7478D8A94A">11.3-6 MinimalPresentation</a></span>
<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"><br /><span class="nocss">  </span><a href="chap11.html#X81CA53DA8216DC82">11.3-9 IsGeneric</a></span>
<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 HomogeneousCatenaryDegreeOfAffineSemigroupUsingexternal packages
<span class="ContSS"><brdiv "ContSSBlock">
< class">br /< nocss"&;nbsp;<spana hrefc.html>11.-10<a>/java.lang.StringIndexOutOfBoundsException: Index 135 out of bounds for length 135
<span class="ContSS"><br /><span class="nocss">  </<span class="ContSS"><br /><span class="nocss">  </span=chap13.tmlX875040237A692C3C"13.1-3NumSgpsUseNormalize<>span
<="chap11.html#>1.412</>/span>
</div></div>
<div span classC">br <span "ocss">nbsp <span>< ".html#X7F7699A9829940C2.15 NumSgpsUseSingularInterface<a<span
      Finitely generated ideals of affine semigroups
  <span>/>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a <span>/>
<span class="ContSS"><br /><span class="nocss">pan =java.lang.StringIndexOutOfBoundsException: Range [22, 19) out of bounds for length 138
<pan=C"</<span =java.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 141
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X8086C1EE7EAAB33D">11.5-4 Generators</a></span/div<div
<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>
</div></div>
</div>
<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="span class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X8307A0597864B098">11.1-21 EquationsOfGroupGeneratedBy
      Defining good semigroups
    </span></a>
</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"</>span ="nocss"&;&bsp;/pan<ahref".#">12.12 NumericalSemigroupDuplication</></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href=/span>
<span class="ContSS"><br /class="ContSS"><br /><span class="nocss">  </span><a href="chap11.html#X7FE3B3C380641DDC">11.2-1 GluingOfAffineSemigroups</a></span>
<=ContSS>/<classnocss">java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 137
</div></div>
<div class=of affine semigroups
       
  <an</>
<>
<divspanjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 150
<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</>/span>
<span =ContSS"> >< =nocss">nbsp <><a href"chap12.XAB83682858A11"12.-5SmallElements<a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html#X82D40159783F0D48"span class=C>br/class=nocss" n;<span>a href"chap11html">36MinimalPresentation/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>
 " >spanclass="n;nbsp< href=chap12html
<span class="ContSS"><br /><span class="nocss">  </<span class="ContSS"><br /><span class="nocss">  </spanhref=chap12html#X7E4FC6DB794992E0">12.5-2 GoodGeneratingSystemOfGoodIdeal</a></span>
 =ContSS>< >span=n"  <span< href".java.lang.StringIndexOutOfBoundsException: Index 146 out of bounds for length 146
span"ContSS>br/<spanclass=nocss">nbsp;n;/<a "html>.-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 class="ContSS"><br /><span class="nocss">  </span href=.#X7DA7AE32837CC1C7">12.5-7 CanonicalIdealOfGoodSemigroup</a></span>
< ContSS< /<span class""&;nbsp;<>< "chap12.#X809E0C077A613806">12.2-15 AperySetOfGoodSemigroup</a></span>
<span class="ContSS"><span =ContSS">br/<span ="" n;/span<ahref=chap12.html#X87AB3B09857B383A">12.5-9 TracksOfGoodSemigroup</a></span>
</div></div>
<div class="<div>
      Symmetric good semigroups
  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap12.html External packages
</div></div>
<class="ContSect"><class=tocline"< nocss"> </span< href=chap12html#X80A3D64386A152EB">12.4 <span class="Heading">
    Arf good closure
  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X8736665E7CBEAB20">13.1-1 NumSgpsUse4ti2</a></span>
</div></div>
<div class="ContSect"><span<span "">< >span""&;nbsp;/><href=chap13htmlX875040237A692C3C">13.1-3 NumSgpsUseNormalize</a></span>
      Good ideals
<java.lang.StringIndexOutOfBoundsException: Range [13, 10) out of bounds for length 15
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class< class=ContChap>< "chap14html#X7B8D661F79E957A6">14 <span class="Heading">
<span class="ContSS"><br /><span class <span></>
spanclass""br/>span class="">&bsp;nbsp<span>a=".html#X82D384397EE5CAC4">12.5-3 AmbientGoodSemigroupOfGoodIdeal</a></span>
<span class="    <span>/a>
<span class="ContSS"><br /><span class="nocss">  </span><a href< class=">
s =ContSS>br/span="nocss> nbsp/><a ="html#X842F3CE07E893949">12.5-6 SmallElements</a></span>
<span class"">b >< class"> nbsp/span< href"chap12#X7DA7AE32837CC1C7">12.5-7 CanonicalIdealOfGoodSemigroup</a></span>
">  </span><a href="chap12.html#X7DC7A4B57BC2E55C">12.5-8 AbsoluteIrreduciblesOfGoodSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss" ;/span>a href="chap12hjava.lang.StringIndexOutOfBoundsException: Range [145, 104) out of bounds for length 145
<div<>
</div>
d class=">ahref=chap13.html#X84A2793F7A9F3E6A">13 <span class="Heading">
 < class""< /<pan="&; /><a href="chap14html#X7F43955582F472B6">14.1-6 DotOverSemigroupsNumericalSemigroup</a></span>
 /</>
<div class< classC"br/span "ocss" &;<span< href="chap14html#X7EC75F477D4F8CC3">14.1-8 DotFactorizationGraph</a></span>
       Using  packages
    <<span=ContSS>br>span =nocss>nbsp;n;java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X8736665E7CBEAB20">13.1-1 NumSgpsUse4ti2</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X875001717A8CF032">13.1-2 NumSgpsUse4ti2gap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X875040237A692C3C">13.1-3 NumSgpsUseNormalize</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7CD12ADD78089CBE">13.1-4 NumSgpsUseSingular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap13.html#X7F7699A9829940C2">13.1-5 NumSgpsUseSingularInterface</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap14.html#X7B8D661F79E957A6">14 <span class="Heading">
  Dot functions
 </span></a>
<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>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X86C990AC7F40E8D0">A.1-2 IsBezoutSequence</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapA.html#X7C9DCBAF825CF7B2">A.1-3 CeilingOfRational</a></span>
</div></div>
<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>
</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>
<java.lang.StringIndexOutOfBoundsException: Index 194 out of bounds for length 194
<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>
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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>
</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>
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<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>
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<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>
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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>
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<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>
</span>
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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>
</span>
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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>
</span>
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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>
</span>
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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>
</span>
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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>
</span>
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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>
</span>
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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>
</span>
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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>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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