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                Presentations of Numerical Semigroups
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<div class="ChapSects"><a href="chap4.html#X7969F7F27AAF0BF1">4 <span class="Heading">
                Presentations of Numerical Semigroups
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X815C0AF17A371E3E">4.1-3 BettiElements</a></span>
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C42DEB68285F2B8">4.1-7 ShadedSetOfElementInNumericalSemigroup</a></span>
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<div java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<="nocss>nbsp;nbsp;<pan>a href="chap4.html#X7C6F554486274CAE">4.3-1 IsUniquelyPresented</a></span>
<span class="ContSS"><br /><span class="div class="func"><table class="func" width%>tr>td class=dleft"<ode ""#8227; MinimalPresentationOfNumericalSemigroup</(<varclass"rg"S/>)td>tdclass=tdright"(nbsp;function&bsp)/td<>java.lang.StringIndexOutOfBoundsException: Range [240, 239) out of bounds for length 246
/>/java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
</div>

<h3>4 <span class="Heading">
                Presentations of Numerical Semigroups
            </span></h3>

<p>In this chapter wethejava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 121

<p>The java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 0

<p><a id="java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0

<h4>4.1 <spanjava.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 92

<>this section we a wayto minimalpresentationsofjava.lang.StringIndexOutOfBoundsException: Range [136, 135) out of bounds for length 337

<p>If the variable <var class="Arg">NumSgpsUseEliminationForMinimalPresentations</var> is set to true, then minimal presentations are computed via binomial ideals and elimination.</p>

<p><a id="X81A2C4317A0BA48D" name="[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0,2]  3,1 0  ] [[0,,1 ,[4 ,  ]] 

<h5>4.[[,2 0,  1 ,1 ]]]

<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 12
< =f"<" "100">tr>tdjava.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 246
<>var class=Arg"S/var> isanumerical semigroup. output  listof java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 568

<p>Any other relation among the minimal generators of the semigroup can be deduced from the ones given in the output.</p>

<p>The algorithm implemented is described in <a href="chapBib.html#biBRos96">[Ros96a]</a> (see also <a href="chapBib.html#biBRGS99">[RG99b]</a>).</p>


<div java.lang.StringIndexOutOfBoundsException: Range [4, 3) out of bounds for length 59
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 92
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0 ] ], 
  [[0 ,0 ,[1 0, ]] java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
<span<>Theis  span=m_1..<istheofjava.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 468
[ [[0,0,2]  ,1 0]], [0,1, 1] [,0,0  , 
  [ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
</pre></div>

<p>The first element in java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0

<p><a id="X81CC5A6C870377E1"

<h5>4.1-2 GraphAssociatedToElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><class=GAPprompt>&;span>span ">(10,);spanjava.lang.StringIndexOutOfBoundsException: Index 120 out of bounds for length 120
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</pre<div>

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

<p>This function is used to compute a minimal presentation of the numerical java.lang.StringIndexOutOfBoundsException: Index 81 out of bounds for length 0


<div class="example"><pre>
<span class="GAPprompt"div class=func">tableclass=""=100%"<<td"dleft> java.lang.StringIndexOutOfBoundsException: Range [87, 85) out of bounds for length 240
<span G>&t;</ span class=GAPinput"GraphAssociatedToElementInNumericalSemigroup,s;<>
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
</pre></div>

<p><a id="X815C0AF17A371E3E" name="X815C0AF17A371E3E"p<var ">Svar> is a semigroup.<

<h5>4.1-3 BettiElements</h5>

<div class=
<class=func"><table class=" width=100%"<tr>td ="tdleft"<code class="func"&8227; java.lang.StringIndexOutOfBoundsException: Range [136, 134) out of bounds for length 240
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>span class="GAPprompt">gap></span> <span class="GAPinput">BettiElementsOfNumericalSemigroup(s);</span>


<div class="example"><pre ,12 14]
<span >a id"7java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<>gap&t</> span ""BettiElementsOfNumericalSemigroups/span>
[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>gapgt</ <span ""BettiElements(s;/java.lang.StringIndexOutOfBoundsException: Index 86 out of bounds for length 86
 10 12 14 java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p><a id="X7FC66A1B82E86FAF" name="X7FC66A1B82E86FAF"></a></p>

<h5>4.1-4 IsMinimalRelationOfNumericalSemigroup</h5>

""tableclass"" "%<>tdclass="tdleft"><code class="func">‣ IsMinimalRelationOfNumericalSemigroup</code>( <var class="Arg">p</var>, <var class="Arg">S</var> )</td><td class="tdright">(&java.lang.StringIndexOutOfBoundsException: Index 229 out of bounds for length 62
<><class"java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 255


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 5
<span class="GAPprompt">gapg;/pan> span class""MinimalPresentation</java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 92
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0
<><span spanclass=GAPinput"IsMinimalRelationOfNumericalSemigroup[210,[,0,2],);/span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalRelationOfNumericalSemigroup
true
</pre>

<p><a id="X8750A6837EF75CA2" div class="func"><table class="func="%>tr><class="tdleft"<codeclass"func>#8227; /(varclass=""Svar>)/td>td""(nbspfjava.lang.StringIndexOutOfBoundsException: Range [216, 215) out of bounds for length 246

<h5>4.1-5 AllMinimalRelationsOfNumericalSemigroup</h5>

<div class
<p><var class="


<div class="example"><pre>
<span classjava.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 99
<java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for length 92
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class=" [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ]
[ [ [ 0, 3, 0 ],[ [ 0 ,0] [ 0,0, 2  ]],   3 0,0]  0 [ 0   ]]
</<<div

<p><a id="X7A9B5AE782CAEA2F" name=p>"java.lang.StringIndexOutOfBoundsException: Range [35, 27) out of bounds for length 62

<h5>4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</h5>

<div class="func"><
<p><var div class="example"><pre>

e istheset java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 468


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreesOfPrimitiveElementsOfNumericalSemigroup( 3, ,7, 10,12,1415,  ]
[ 3, 5,
</pre><div>

<ph5>417ShadedSetOfElementInNumericalSemigroup</h5>

<h5>4.1-7 ShadedSetOfElementInNumericalSemigroup</h5>

<div class="func"><table <<class=A"S/ is  numerical and <class="Arg">n</var> is an element in <var class="Arg">S</var>.</p>
<p>ar class=A"S/var>is a java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 128

<p>The output

<p>java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0


<div class="example"><pre>
<class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</spanjava.lang.StringIndexOutOfBoundsException: Index 98 out of bounds for length 39
<> =java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 114
[ [  ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
</pre></div>

<p><a id="X795E7F5682A6C8B3" name="java.lang.StringIndexOutOfBoundsException: Index 38 out of bounds for length 0

<h4>4.2 <span class="Heading">Binomial ideals associated to java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 0

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

<pclass=func"><table class="func" width="100%"><tr><td class="tdleft"><p>The argument <var class="Arg">K</var> is optional; when it is not supplied,  java.lang.StringIndexOutOfBoundsException: Range [109, 108) out of bounds for length 260

<p><a id="X7E6BBAA7803DE7F3" name="X7E6BBAA7803DE7F3"></a></p>

<h5>4.2-1 java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><span class="java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 99
<p>The argument <var class="Arg">K</var> is optional; when it is not supplied, the span class="GAPprompt">gap></span> <span class="GAPinput">BinomialIdealOfNumericalSemigroup>


<example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class=span ""gapg;<span>span ">BinomialIdealOfNumericalSemigroups;span>
&two-ideal GF()x_1,_2,3] ( &;
<span class="GAPpromptspan class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[ x_1^*x_2+x_3^2,x_1^4x_2*, *x_3+x_2^2]
<span class="GAPprompt">gap></span> <span class="GAPinput">BinomialIdealOfNumericalSemigroup "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 92
<two-sided ideal in Rationals[x_1,x_2, [ [, ,2]  3 1,0]] [  0, 1 1], [4,0,  ]],
<span class="[[ 0,2, 0],[ 1, 0 1 ]] 
[ -x_1^3*x_2+x_3^2, -x_1^4+x_2*x_3, -x_1*x_3+x_2^2 ]
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0
[ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
</pre></div>

<p><a id="X7D7EA20F818A5994" name="X7D7EA20F818A5994"></a></p>

<h4>4.3 <span class="Heading">Uniquely Presented Numerical

<p>A numerical semigroup <span class="SimpleMath">S</span> is uniquely<iv class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classp<var class"Arg"S<var>is numerical semigroup</java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57

<p><a id="X7C6F554486274CAE" 

<h5>4.3-1 IsUniquelyPresented</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsUniquelyPresentedspan ""gap><span<G>(s<>
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 244
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>The output is true if <var class="Arg">S</var> has uniquely presented. The implementation is based on <a href="chapBib.html#biBGS-O">[GO10]</a>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap&h5>32 java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 24
<pan class="GAPprompt">gap></span> <span class="GAPinput">IsUniquelyPresented(s);</span>
true
<span class="GAPprompt">gap></span> <span<iv class=">< class"unc" java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 234
true
</pre></div>

<p><a id="X79C010537C838154" name="X79C010537C838154"></a></p>

<h5>4.3-2 IsGeneric</h5>

<ivclass="func><class= width"%>trtdtjava.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 216
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsGenericNumericalSemigroup</code>( <var class="Arg">S</var> )</td><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><var class="Arg">S</var> is a numerical java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 0

<p>The output is true if <var class="Arg">S</var> has a generic presentation, that is, in every minimal relation all java.lang.StringIndexOutOfBoundsException: Range [0, 127) out of bounds for length 0

<p>This filter implies <code class="func">IsUniquelyPresentedNumericalSemigroup</code> (<a href="chap4.html#X7C6F554486274CAE"><span class="RefLink"<panclass=G"gapgt;/span> <span class="GAPinput">IsGeneric(s);</span>


<div class/pre>/
<span class="GAPprompt
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
rue
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGenericNumericalSemigroup(s);</span>
true
</pre></div>


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