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<div class="ChapSects"><a href="chap4_mj.html#X7969F7F27AAF0BF1">4 <span class="Heading">
                Presentations of Numerical Semigroups
            </span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7969F7F27AAF0BF1">4.1 <span class="Heading">Presentations of Numerical Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81A2C4317A0BA48D">4.1-1 MinimalPresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X81CC5A6C870377E1">4.1-2 GraphAssociatedToElementInNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X815C0AF17A371E3E">4.1-3 BettiElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7FC66A1B82E86FAF">4.1-4 IsMinimalRelationOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8750A6837EF75CA2">4.1-5 AllMinimalRelationsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7A9B5AE782CAEA2F">4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7C42DEB68285F2B8">4.1-7 ShadedSetOfElementInNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.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_mj.html#X7E6BBAA7803DE7F3">4.2-1 BinomialIdealOfNumericalSemigroup</a></span>
</div></div>
<div class="ContSect"><span class<<a id"X7969F7F27AAF0BF1 =X7969F7F27AAF0BF1">/</p>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a hrefPresentations of Semigroups
<java.lang.StringIndexOutOfBoundsException: Range [27, 5) out of bounds for length 134
</div>/div>
</div>

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

<p>In this chapter we explain how to compute a minimal presentation of a <class=""br /< class=nocss>;&bsp;/>< hrefspan>

<p>The set of minimal generators is stored in"&;nbsp;</span>a ="hap4_mjhtml#X815C0AF17A371E3E>.1-3  BettiElements/a>/pan

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

<h4>4.1 <span class="Heading">Presentations of Numerical Semigroups</span></h4>

<p>In this section we provide a way to compute minimal presentations of a numerical semigroup. These presentations are constructed from some special elelements in the semigroup (Betti <span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8750A6837EF75CA2">4.1-5 AllMinimalRelationsOfNumericalSemigroup</a></span>

<p>If the variable <var class="Arg">NumSgpsUseEliminationForMinimalPresentations</var> is set to true, then minimal presentations are computed </></div

<p><a id="X81A2C4317A0BA48D" name="X81A2C4317A0BA48D"></a></p>

<h5>4.1-1 MinimalPresentation</h5><span

<div class="func"><table class="func" width="100%"><></div
<class=func"<table class=func"width=100%tr< class"dleft"< ="func"># </code(< classArg>S<> )</>td "tdright> &bsp;<td></></able>/>
<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/span

<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_mj.html#biBRos96">[Ros96a]</a> (see also <a href=span class=""<br/<spanclass""&;n;/>< hrefchap4_mj.X79C010537C838154"4.32 </a>/span

<div class="example"><pre>
<p this   howto   of numerical.   a  presentationis   java.lang.StringIndexOutOfBoundsException: Range [153, 152) out of bounds for length 682
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0 ] ],
  [ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentationOfNumericalSemigroup(s);</span>
[ [ [ 0, 0, 2 ], [ 3, 1, 0 ] ], [ [ 0, 1, 1 ], [ 4, 0, 0 ] ],
   0 2, 0][1, , ]  java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
</pre></div>

<> firstin listmeansthatspan ">(1times31 72times5 \/java.lang.StringIndexOutOfBoundsException: Range [111, 110) out of bounds for length 154

<p><a id="X81CC5A6C870377E1" name="X81CC5A6C870377E1"></a></p>

<h5>4.1-2 GraphAssociatedToElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class
numerical semigroup <var class="rg"n/>isan  java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 128

<puta   < ="({_1\java.lang.StringIndexOutOfBoundsException: Range [69, 68) out of bounds for length 505

<p>This function is used to compute a minimal presentation of the numerical semigroup <var class="Arg">S</var>, as explained in <a href="chapBib_mj.html#div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalPresentationOfNumericalSemigroup</code>( <var class>td >< class"">&8227 java.lang.StringIndexOutOfBoundsException: Range [141, 140) out of bounds for length 246

<div class="example"><pre>
<> <span class=":=java.lang.StringIndexOutOfBoundsException: Range [84, 83) out of bounds for length 99
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ [ 3, 5, 7 ], [ [ 3, 7
</pre></div>

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

<h5>4. 0 ][[0 ,1 ][4 0  ],

<div class="func"><table class="func" width="100%"><tr><td    ,,0], 1  ]] java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32
<div; BettiElementsOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<<=Arg"S</> is a  ./p>

<p>The output is the set of elements in <var class="Arg">S</var> whose associated graph is nonconnected <a href=  [ 0 ,0]  ,0   java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 32

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<class"">apgt;/> class=>java.lang.StringIndexOutOfBoundsException: Range [96, 95) out of bounds for length 106
[ 10, 12, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BettiElements(s);</span>
[10, 12 14]
</pre></div>

<p><a id="X7FC66A1B82E86FAF"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<h5>4.1-4 IsMinimalRelationOfNumericalSemigroup</h5>

<div class="func
<p><var class=span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 99

<div class="example"><pre>
<> <span ="java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 99
<span class="GAPprompt">gap></span>[ 3,5,7] [ [ [ 3,7]]]
[[ [ , 0,2,[0 ,0 ] [ , ,  ]  ,0 0]]]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalRelationOfNumericalSemigroup([[2,1,0],java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalRelationOfNumericalSemigroup([[3,1,0],[0,0,2]],s);</span>
true
</pre></div>

<p><a id="X8750A6837EF75CA2" name="X8750A6837EF75CA2"></a></p>

<h5>4.1-5 AllMinimalRelationsOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; AllMinimalRelationsOfNumericalSemigroup</code>( <varp<var ="S>  anumerical</>
< presentations of <var class="S<var>. that [y]  a  either y] [y, be in     both.p

<div class="example"><pre>
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 14
<span class="GAPprompt">gap></span> <span class="GAPinput">[10 ,
[], [ 3, 0, 0 ] ] java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
<">gap></>span GAPinput>java.lang.StringIndexOutOfBoundsException: Range [102, 101) out of bounds for length 112
[ [ [ 0, 3, 0 ], [
</></java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

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

<h5>4.1-6 DegreesOfPrimitiveElementsOfNumericalSemigroup</h5>span =Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 92

<div span clas="APprompt"gt;/span> <span class="GAPinput"IsMinimalRelationOfNumericalSemigroup[2,1,][00,]],)span>
<p ""S>is  <p

<p>The output is the java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 4

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 0
ss"">java.lang.StringIndexOutOfBoundsException: Range [109, 108) out of bounds for length 119
[ 3, 5, 7, 10, 12p><classArg><var>is numerical    isunion  java.lang.StringIndexOutOfBoundsException: Range [93, 92) out of bounds for length 246
</pre></div>

<p><a id="X7C42DEB68285F2B8" name="X7C42DEB68285F2B8"></a></p>

<h5>4.1-7 ShadedSetOfElementInNumericalSemigroup</h5>

<class=%>tr<class=""< class=""&8227;java.lang.StringIndexOutOfBoundsException: Range [140, 139) out of bounds for length 271
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> is an element in <var class="Arg">S</var>.</p>

<p>The output is a simplicial complex <span class="SimpleMath">\(C\)</span>. If <pan java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 99

<is function  aof graph<var"rg"><.<p

<"<>
<span class="[  [0 3,0],  ,0,] [ [3, ,0]  0,2,0] ,[[3 ,] 0 0,2  ]]
<span class="GAPprompt">gap></span> <span class="GAPinput">ShadedSetOfElementInNumericalSemigroup(10,s);</span>
[  ,  3 ]  37 ,[ ,[7  java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 39
</pre></div>

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

<p> =""Sisasemigroupp

<>Let<span">(\<span>  a , and <spanclass"impleMath>\(\</pan> field Let< class=>\\,dots,})/span>be  ofminimalgenerators of < =>\(\)/>  let span =>\K,\ots,]\</>be ringof  theindeterminates <class"SimpleMath">\(x_1,\dots,x_e\)</span> and with coefficients in <span class="SimpleMath">\(K\)</span>. Let <span class="SimpleMath">\(K[t]\)</span> be the ring of polynomials in <span class="SimpleMath">\(t\)</span> with coefficients in <span class="SimpleMath">\(K\)</span>.</p>

<p>Let <span class="SimpleMath">\(\varphi: K[x_1,\dots,x_e] \to K[t]\)</span> be the ring homomorphism determined by <span class="SimpleMath">\(\varphi(x_i)=t^{n_i}\)</span> for all java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>a id=X7name="java.lang.StringIndexOutOfBoundsException: Range [56, 52) out of bounds for length 62

<h5>4.2-1 BinomialIdealOfNumericalSemigroup</h5>

<p =X7java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<p> var java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 260

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5td =tdleft><=func>#java.lang.StringIndexOutOfBoundsException: Range [141, 139) out of bounds for length 271
<span class="GAPprompt">gap></span> <span class="GAPinput">BinomialIdealOfNumericalSemigroup(GF(2),s);</span>
<two-sided ideal in GF(2)[x_1,x_2,x_3], (3 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[x_13+x_32 ^4* *x_3x_2^ java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<span class="GAPprompt">gap></span> <span class="java.lang.StringIndexOutOfBoundsException: Index 54 out of bounds for length 0
&; in [,x_2,3](3 &;
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[x_1^3*+_^,-_^+x_2x_3, -*java.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 52
<span class=" class="GAPprompt;span<span=java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 99
[   , , [, ,  ]] [[1, 1,[4 , ]
[ [ 0, 2, 0 ], [ 1, 0, 1 ] ] ]
<

<p>p<id"java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length 0

<p>A numerical semigroup <span class="SimpleMath">\(S\)</span> is uniquely presented

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

<h5>4.3-1 IsUniquelyPresented</h5>

<div class="func">p>Let <span ="SimpleMath">\S\)>  a    span ="">K)/span  field<class"java.lang.StringIndexOutOfBoundsException: Range [157, 156) out of bounds for length 659
<class""< =func width"%>trtd class="tdleft"><code class="func">‣ IsUniquelyPresentedNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<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_mj.html#biBGS

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);
<

<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
true
</pre></div>

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

<h5>4.3-2 IsGeneric</h5>

<java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0
<div class="func"< =GAPprompt&;</java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 99
<p><var class="Arg">S</var> is a numerical semigroup.</p>

<p>The output is true if <var class="ltt-sided ()x1,_2,_33,( java.lang.StringIndexOutOfBoundsException: Range [58, 56) out of bounds for length 61

<p>This filter implies <code [x_1^x_324+ *x_22 ]

<div class="example"><pre>
<span&;sidedjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 65
<="GAPinput"
true
<span class=-3*x_2x_32,-x_1^+java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 52
true
</pre></div>

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