Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/numericalsgps/doc/   (Algebra von RWTH Aachen Version 4.15.1©)  Datei vom 30.7.2024 mit Größe 22 kB image not shown  

Quelle  chap4_mj.html   Sprache: HTML

 
 products/Sources/formale Sprachen/GAP/pkg/numericalsgps/doc/chap4_mj.html


<?xml version="1.0" encoding="UTF-8"?>

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
         "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (NumericalSgps) - Chapter 4: 
                Presentations of Numerical Semigroups
            </title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap4"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chap14_mj.html">14</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapC_mj.html">C</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap3_mj.html">[Previous Chapter]</a>    <a href="chap5_mj.html">[Next Chapter]</a>   </div>

<p id="mathjaxlink" class="pcenter"><a href="chap4.html">[MathJax off]</a></p>
<p><a id="X7969F7F27AAF0BF1" name="X7969F7F27AAF0BF1"></a></p>
<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="tocline"><span class="nocss"> </span><a href="chap4_mj.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_mj.html#X7C6F554486274CAE">4.3-1 IsUniquelyPresented</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X79C010537C838154">4.3-2 IsGeneric</a></span>
</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 numerical semigroup. Recall that a minimal presentation is a minimal generating system of the kernel congruence of the factorization map of the numerical semigroup. If <span class="SimpleMath">\(S\)</span> is a numerical semigroup minimally generated by <span class="SimpleMath">\(\{n_1,\ldots,n_e\}\)</span>, then the factorization map is the epimorphism <span class="SimpleMath">\(\varphi: \mathbb{N}^e\to S\)</span>, <span class="SimpleMath">\((x_1,\ldots,x_e)\mapsto x_1n_1+\dots+ x_en_e\)</span>; its kernel is the congruence <span class="SimpleMath">\(\{ (a,b) \mid \varphi(a)=\varphi(b)\}\)</span>.</p>

<p>The set of minimal generators is stored in a set, and so it may not be arranged as the user gave them. This may affect the arrangement of the coordinates of the pairs in a minimal presentation, since every coordinate is associated to a minimal generator.</p>

<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 elemenents) whose associated graphs are nonconnected. A generalization of these graphs are the simplicial complexes called shaded sets of an element.</p>

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

<h5>4.1-1 MinimalPresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalPresentation</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalPresentationOfNumericalSemigroup</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. 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">\( \{ \{x_1,y_1\},\ldots,\{x_k,y_k\}\} \)</span> is the output and <span class="SimpleMath">\( \{m_1,\ldots,m_n\} \)</span> is the minimal system of generators of the numerical semigroup, then <span class="SimpleMath">\( \{x_i,y_i\}=\{\{a_{i_1},\ldots,a_{i_n}\},\{b_{i_1},\ldots,b_{i_n}\}\}\)</span> and <span class="SimpleMath">\( a_{i_1}m_1+\cdots+a_{i_n}m_n= b_{i_1}m_1+ \cdots +b_{i_n}m_n.\)</span></p>

<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="chapBib_mj.html#biBRGS99">[RG99b]</a>).</p>


<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">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, 0, 1 ] ] ]
</pre></div>

<p>The first element in the list means that <span class="SimpleMath">\( 1\times 3+1\times 7=2\times 5 \)</span>, and the others have similar meanings.</p>

<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="func">‣ GraphAssociatedToElementInNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<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 pair. If <span class="SimpleMath">\( \{m_1,\ldots,m_n\} \)</span> is the set of minimal generators of <var class="Arg">S</var>, then the first component is the set of vertices of the graph associated to <var class="Arg">n</var> in <var class="Arg">S</var>, that is, the set <span class="SimpleMath">\(\{ m_i \ |\ n-m_i\in S\} \)</span>, and the second component is the set of edges of this graph, that is, <span class="SimpleMath">\( \{ \{m_i,m_j\} \ |\ n-(m_i+m_j)\in S\}.\)</span></p>

<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#biBRos96">[Ros96a]</a>.</p>


<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">GraphAssociatedToElementInNumericalSemigroup(10,s);</span>
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
</pre></div>

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

<h5>4.1-3 BettiElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BettiElements</code>( <var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BettiElementsOfNumericalSemigroup</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 in <var class="Arg">S</var> whose associated graph is nonconnected <a href="chapBib_mj.html#biBGS-O">[GO10]</a>.</p>


<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">BettiElementsOfNumericalSemigroup(s);</span>
[ 10, 12, 14 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">BettiElements(s);</span>
[ 10, 12, 14 ]
</pre></div>

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

<h5>4.1-4 IsMinimalRelationOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMinimalRelationOfNumericalSemigroup</code>( <var class="Arg">p</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">p</var> is a pair (a relation) of lists of integers. Determines if the pair <var class="Arg">p</var> is a minimal relation in a minimal presentation of <var class="Arg">S</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(4,6,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMinimalRelationOfNumericalSemigroup([[2,1,0],[0,0,2]],s);</span>
false
<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">‣ AllMinimalRelationsOfNumericalSemigroup</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. The output is the union of all minimal presentations of <var class="Arg">S</var>. Notice that if [x,y] is a minimal relator, then either [x,y] or [y,x] will be in the output, but not both.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(4,6,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalPresentation(s);</span>
[ [ [ 0, 0, 2 ], [ 0, 3, 0 ] ], [ [ 0, 2, 0 ], [ 3, 0, 0 ] ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllMinimalRelationsOfNumericalSemigroup(s);</span>
[ [ [ 0, 3, 0 ], [ 0, 0, 2 ] ], [ [ 3, 0, 0 ], [ 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">‣ 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 <span class="SimpleMath">\(s\)</span> in <var class="Arg">S</var> such that there exists a minimal solution to <span class="SimpleMath">\(msg\cdot x-msg\cdot y = 0\)</span>, such that <span class="SimpleMath">\(x,y\)</span> are factorizations of <span class="SimpleMath">\(s\)</span>, and <span class="SimpleMath">\(msg\)</span> is the minimal generating system of <var class="Arg">S</var>. Betti elements are primitive, but not the way around in general.</p>


<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(s);</span>
[ 3, 5, 7, 10, 12, 14, 15, 21, 28, 35 ]
</pre></div>

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

<h5>4.1-7 ShadedSetOfElementInNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ShadedSetOfElementInNumericalSemigroup</code>( <var class="Arg">n</var>, <var class="Arg">S</var> )</td><td class="tdright">( function )</td></tr></table></div>
<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 <span class="SimpleMath">\( \{m_1,\ldots,m_n\} \)</span> is the set of minimal generators of <var class="Arg">S</var>, then <span class="SimpleMath">\(L \in C\)</span> if <span class="SimpleMath">\(n-\sum_{i\in L} m_i\in S\)</span> (<a href="chapBib_mj.html#biBSzW">[SW86]</a>).</p>

<p>This function is a generalization of the graph associated to <var class="Arg">n</var>.</p>


<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">ShadedSetOfElementInNumericalSemigroup(10,s);</span>
[ [  ], [ 3 ], [ 3, 7 ], [ 5 ], [ 7 ] ]
</pre></div>

<p><a id="X795E7F5682A6C8B3" name="X795E7F5682A6C8B3"></a></p>

<h4>4.2 <span class="Heading">Binomial ideals associated to numerical semigroups</span></h4>

<p>Let <span class="SimpleMath">\(S\)</span> be a numerical semigroup, and let <span class="SimpleMath">\(K\)</span> be a field. Let <span class="SimpleMath">\(\{n_1,\dots,n_e\}\)</span> be a set of minimal generators of <span class="SimpleMath">\(S\)</span>, and let <span class="SimpleMath">\(K[x_1,\dots,x_e]\)</span> be the ring of polynomial in the indeterminates <span 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 <span class="SimpleMath">\(i\)</span>. The image of this morphism is usually known as the <em>semigroup ring associated</em> to <span class="SimpleMath">\(S\)</span>. The kernel is the <em>(binomial) ideal associated</em> to <span class="SimpleMath">\(S\)</span>. According to <a href="chapBib_mj.html#biBMR0269762">[Her70]</a>, from the exponents of the binomials in this ideal we can recover a presentation of <span class="SimpleMath">\(S\)</span> and viceversa.</p>

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

<h5>4.2-1 BinomialIdealOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BinomialIdealOfNumericalSemigroup</code>( [<var class="Arg">K</var>, ]<var class="Arg">S</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>The argument <var class="Arg">K</var> is optional; when it is not supplied, 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>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<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_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></span> <span class="GAPinput">BinomialIdealOfNumericalSemigroup(s);</span>
<two-sided ideal in Rationals[x_1,x_2,x_3], (3 generators)>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfTwoSidedIdeal(last);</span>
[ -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></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 ] ] ]
</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 class="SimpleMath">\(S\)</span> is uniquely presented if for any two minimal presentations <span class="SimpleMath">\(\sigma\)</span> and <span class="SimpleMath">\(\tau\)</span> and any <span class="SimpleMath">\((a,b)\in \sigma\)</span>, either <span class="SimpleMath">\((a,b)\in \tau\)</span> or <span class="SimpleMath">\((b,a)\in \tau\)</span>, that is, there is essentially a unique minimal presentation (up to arrangement of the components of the pairs in it).</p>

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

<h5>4.3-1 IsUniquelyPresented</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsUniquelyPresented</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td 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-O">[GO10]</a>.</p>


<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">IsUniquelyPresented(s);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsUniquelyPresentedNumericalSemigroup(s);</span>
true
</pre></div>

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

<h5>4.3-2 IsGeneric</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGeneric</code>( <var class="Arg">S</var> )</td><td class="tdright">( property )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsGenericNumericalSemigroup</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 a generic presentation, that is, in every minimal relation all generators occur. These semigroups are uniquely presented (see <a href="chapBib_mj.html#biBB-GS-G">[BGG11]</a>).</p>

<p>This filter implies <code class="func">IsUniquelyPresentedNumericalSemigroup</code> (<a href="chap4_mj.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>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGeneric(s);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsGenericNumericalSemigroup(s);</span>
true
</pre></div>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap3_mj.html">[Previous Chapter]</a>    <a href="chap5_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chap9_mj.html">9</a>  <a href="chap10_mj.html">10</a>  <a href="chap11_mj.html">11</a>  <a href="chap12_mj.html">12</a>  <a href="chap13_mj.html">13</a>  <a href="chap14_mj.html">14</a>  <a href="chapA_mj.html">A</a>  <a href="chapB_mj.html">B</a>  <a href="chapC_mj.html">C</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>

100%


¤ Dauer der Verarbeitung: 0.16 Sekunden  (vorverarbeitet)  ¤

*© Formatika GbR, Deutschland




Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.