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<title>GAP (NumericalSgps) - Chapter 4: 
                Presentations of Numerical Semigroups
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81A2C4317A0BA48D">4.1-1 MinimalPresentation</a></span>
<java.lang.StringIndexOutOfBoundsException: Range [52, 5) out of bounds for length 166
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#java.lang.StringIndexOutOfBoundsException: Index 101 out of bounds for length 56
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< "">br/span"nocss" nbsp;<span< "h#8750">.15</a</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A9B5AE782CAEA2F">4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</a></span>
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<h3>4 <span class="Heading">
                Presentations of Numerical Semigroups
            </span></h3>

<p>In this chapter we explainspan

<p>The set of minimal generators is stored in a set, and so it mayspan class="ContSS"><br /><span class="nocssspan class"ContSS></<span"ocss&;bsp</span>< ".#81CC5A6C870377E1>.12 GraphAssociatedToElementInNumericalSemigroupa/>

<p><a id="X7969F7F27AAF0BF1" name="X7969F7F27AAF0BF1"></a></p>

<h4>4.1span class="ContSS"><br /><span class="nocss">  <span>a =".X7A9B5AE782CAEA2F"4.6</a>/java.lang.StringIndexOutOfBoundsException: Range [168, 167) out of bounds for length 168

<p>In /div>div>

<p>If the variable <var class="Arg">NumSgpsUseEliminationForMinimalPresentations</var> is set to true

<p><<span class="ContSS"><br /><span class="nocss">  </span></

<h5>4.1-1 MinimalPresentation</h5>

<div class="func"><table class="span class="ContSS"><br /><span class" &;</> java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds for length 141
<="100%<<tdclass="><odeclass=func"‣ /ode>(<var class="rg"</var </td<td class"tdright" function&bsp;</><tr</table></div>
<p><var class="Arg">S</var> is a numerical semigroup. The output is a list of lists with two elements. Each list of two elements represents a relation between the minimal generators of the numerical semigroup. If <span class="SimpleMath">{ {x_1,y_1},...,{x_k,y_k}}</span> is the output and <span class="SimpleMath">{m_1,...,m_n}</span> is the minimal system of generators of</iv<div>

<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]</


<div class="example"><pre>
<span class="
<GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0
  <In this section provide a way compute   of a numerical semigroup. These presentations are constructed from some special elelements in the semigroup (Betti elemenents) whose associated graphs are nonconnected. A generalization of these graphs are the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentationOfNumericalSemigroup(s);</
[ [ [ 0, 0,  ,[ 3 ,0 ]],[  0,1 1]  ,00] , 
  [ [ 0, 2,0  ] [1,0 1  ]]
</pre></div>

<p>The first element in the list means that <span class="SimpleMath">1× 3+1× 7=2divclass"unc>table class=func"width=100%><tr< class="tdleft"><code class="func">&#8227; MinimalPresentationOfNumericalSemigroup</code>( <var class="Arg">p< "><var>   numerical .The isa  oflists with two elements. Each list of two elements represents a relationjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<p

<h5>4.1-2 GraphAssociatedToElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code classspan class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
<p><var class="Arg">S</   [,2 0] [,0,1  ]]

<p output is a pair. If<span class"SimpleMath">{m_1,...,m_n}/span> is the set of minimal generators of <var class="Arg">S</var>, then the first component is  [[ 0 0  ][3 ,0  ]], [ [ 0, 1, 1 , [4 0 0]], 

<p>This function is used to compute a minimal presentation of the


<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span ""gapgt</ <class=GAPinput"GraphAssociatedToElementInNumericalSemigroup,s;<span>
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
</>/

<p><a id="X815C0AF17A371E3E" name="X815C0AF17A371E3E"></a></p>

<h5>4.1-3 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0

<div class="func"><table class="func" width="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< classclass""< "func" width=100%>tr>td class=t"<code class="func">‣ BettiElementsOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( class="APprompt"gap&t;<span><span class"GAPinput>(10,)<span
<>var class="Arg"></ is anumerical semigroup/p>

<p>The output is the set of elements in <var class="Arg">S</var> whose associated graph is nonconnected <a href


<div class""<tableclass"unc" "%">tr<classtdleft>codeclass"func>#8227;BettiElementsOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<span
<java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for length 106
[ 10, 12, 14 ]
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0
[ 10, 12,14 ]
</pre></div>

<p<a id="7FC66A1B82E86FAF" name="X7FC66A1B82E86FAF"></a></p>

<h5>4.1-4 span class="GAPprompt"&;span <class=GAPinput">();</span

<div class="func"><span class="GAPprompt"&;/pan><span class="APinput">();/span>
<p><var class=10, ,14 ]


<div class="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 0
<span <span classfunc">< class=func"width=100">tr< =java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 270
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]
<span class="p<ar =Arg">S</var> is a numerical semigroup and <var class="Arg">p</var> is
false
<span class="GAPprompt">gap></span> <span class="GAPinput">span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);&t;</ <spanclass=GAPinput"(s);<span>
java.lang.StringIndexOutOfBoundsException: Range [56, 57) out of bounds for length 4
</pre></div>

<p><a id="span class="GAPprompt">gap;</><span "GAPinput"([2,1,],[,02],;<java.lang.StringIndexOutOfBoundsException: Range [128, 127) out of bounds for length 128

<h5>4.1-5 AllMinimalRelationsOfNumericalSemigroup</h5>

<" width=100"<tr>td "tdleft"code class=func"‣AllMinimalRelationsOfNumericalSemigroup<code>( < "rg></var </td<td class="tdright">(nbsp;unction )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 0


<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 62
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllMinimalRelationsOfNumericalSemigroup(s);</span>
[[,3,0 ][ 00,, 2]],[ ,0  ,[ 0, 2, 0,2, 0 ] ], [ [ 3, 1, 0 ],  0, 0,2]  ]
<pre>/div>

<p>a id="X7A9B5AE782CAEA2F" name="X7A9B5AE782CAEA2F"></a></p>

<h5>4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; DegreesOfPrimitiveElementsOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>The output is the set of elements <java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 0


<java.lang.StringIndexOutOfBoundsException: Range [19, 4) out of bounds for length 26
<span outputis the set ofelements <span class="SimpleMath">s</span> in <var class="Arg">S</var> such that there exists a minimal solution to <span class="SimpleMath">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">DegreesOfPrimitiveElementsOfNumericalSemigroup(s);</span>
[3,5, 7, 10, 12 14, 15, 21,28,35 ]
</pre></div>

<p/</

<4.- java.lang.StringIndexOutOfBoundsException: Range [52, 48) out of bounds for length 53

java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>var "rg"<var>isanumericalsemigroup  var java.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 128

<p>< "rg><var is a numerical semigroup and <var class="Arg">n</var> is an element in <var class="Arg">S</var>.</p>

<p>This


<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
<spanspan java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 99
[ [  ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
</pre></div>

<p><a id="X795E7F5682A6C8B3" name="span class="GAPprompt">gap></span<span class=GAPinput">ShadedSetOfElementInNumericalSemigroup(10,s);</span>

<h4>4.2 <span class="Heading">Binomial ideals associated to numerical semigroups</java.lang.StringIndexOutOfBoundsException: Index 84 out of bounds for length 12

<p>Let <span class="SimpleMath">S</span> be a numerical semigroup, and let <span class="SimpleMath">K<

<p>Let <span class="SimpleMath">φ: K

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

<h5>4.2-1 BinomialIdealOfNumericalSemigroup</h5>

<div "java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 269
<the field of rational numbers is taken as base field. <var class="Arg">S</var> is a numerical semigroup. The output is the binomial ideal associated to <var class="Arg">S</var>.</p>


<div class="example"><pre>
< =GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<(GF(2),s);</span
<two-sided ideal in GF(2)[x_1,x_2,x_3], (3 generators)>
<span classjava.lang.StringIndexOutOfBoundsException: Range [25, 19) out of bounds for length 26
[ x_1^3*x_2+x_3^2, x_1^4+x_2*x_3, x_1*x_3+x_2^2 ]
<class=GAPprompt>&t;</span <class=GAPinput"BinomialIdealOfNumericalSemigroup()</java.lang.StringIndexOutOfBoundsException: Index 106 out of bounds for length 106
<two<sided idealin 2[x_1,2,_3, 3generators)gt;
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 101
[ -x_1^3*x_2+x_3^2, -x_1^4+x_2*x_3, -x_1* 3x_2+2 ^+*x_3 x_1x_3+x_22 
<spanclass="APprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[[[0,0  ,[3,1   ,[[0,,1 , [4 , 0]], 
  0,2, 0 ,[ 1 ,1 ]]
</pre></div>

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

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

<p>A numerical semigroup <span java.lang.StringIndexOutOfBoundsException: Range [1, 36) out of bounds for length 12

<p><a id="X7C6F554486274CAE" name="X7C6F554486274CAE"></a

<h5>4.3-1 IsUniquelyPresented</h5>

<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 244
<>var="Arg>/  a 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
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 0
<spanclass=GAPprompt">&;<span> <span class="APinput"IsUniquelyPresenteds);<spanjava.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for length 92
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniquelyPresentedNumericalSemigroup(s);</spandiv class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutOfBoundsException: Index 67 out of bounds for length 57
true
</pre></div>

<p><a id="X79C010537C838154"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>43-2IsGeneric</h5>

<div class="func"><table class="func" width="100%"><tr><td <pan
< ="func"<table="unc" width="100%"><tr><td class="tdleft"><code class="func">&#8227; IsGenericNumericalSemigroup</code>( <var class="Arg">S</var> )</java.lang.StringIndexOutOfBoundsException: Range [0, 167) out of bounds for length 4
< "func">able =func"width="100"<tr><td class="dleft"><code class="func">‣ IsGeneric</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>

<p>The output is true if <var class="Arg">S</var> has a generic presentation, that is, in every minimal relation all generators occur. These semigroups are uniquely presented (see <a href="chapBib.html#biBB-GS-G">[BGG11]</a>).</p>

<p>This filter implies <code class="func">IsUniquelyPresentedNumericalSemigroup</code> (<a href="chap4.html#X7C6F554486274CAE"><span class="RefLink">4.3-1</span></a>).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
< class"APprompt">&;span java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 82
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGenericNumericalSemigroup(s);</span>
true
</>div>


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