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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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<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7969F7F27AAF0BF1">4.1 <span class="Heading">Presentations of Numerical Semigroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8028208F84950722">4.1-1 FortenTruncatedNCForNumericalSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D8199858781EB41">4.1-2 MinimalPresentationOfNumericalSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X81CC5A6C870377E1">4.1-3 GraphAssociatedToElementInNumericalSemigroup</a></span>
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<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. There are three functions involved in this process.</p>
<p><a id="X7969F7F27AAF0BF1" name="X7969F7F27AAF0BF1"></a></p>
<h4>4.1 <span class="Heading">Presentations of Numerical Semigroups</span></h4>
<p><a id="X8028208F84950722" name="X8028208F84950722"></a></p>
<h5>4.1-1 FortenTruncatedNCForNumericalSemigroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> FortenTruncatedNCForNumericalSemigroups</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">L</var> contains the list of coefficients of a single linear equation. This function gives a minimal generator of the affine semigroup of nonnegative solutions of this equation with the first coordinate equal to one (see <a href="chapBib.html#biBMR1283022">[CD94]</a>). Returns <code class="code">fail</code> if no solution exists.</p>
<table class="example">
<tr><td><pre>
gap> FortenTruncatedNCForNumericalSemigroups([ -57, 3 ]);
[ 1, 19 ]
gap> FortenTruncatedNCForNumericalSemigroups([ -57, 33 ]);
fail
gap> FortenTruncatedNCForNumericalSemigroups([ -57, 19 ]);
[ 1, 3 ]
</pre></td></tr></table>
<p><a id="X7D8199858781EB41" name="X7D8199858781EB41"></a></p>
<h5>4.1-2 MinimalPresentationOfNumericalSemigroup</h5>
<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 { {x_1,y_1},...,{x_k,y_k}} is the output and {m_1,...,m_n} is the minimal system of generators of the numerical semigroup, then {x_i,y_i}={{a_i_1,...,a_i_n},{b_i_1,...,b_i_n}} and a_i_1m_1+cdots+a_i_nm_n= b_i_1m_1+ cdots +b_i_nm_n.</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.html#biBRos96">[Ros96a]</a> (see also <a href="chapBib.html#biBRGS99">[RG99]</a>).</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> MinimalPresentationOfNumericalSemigroup(s);
[ [ [ 1, 0, 1 ], [ 0, 2, 0 ] ], [ [ 4, 0, 0 ], [ 0, 1, 1 ] ],
[ [ 3, 1, 0 ], [ 0, 0, 2 ] ] ]
</pre></td></tr></table>
<p>The first element in the list means that 1x 3+1x 7=2x 5, and so on.</p>
<p><a id="X81CC5A6C870377E1" name="X81CC5A6C870377E1"></a></p>
<h5>4.1-3 GraphAssociatedToElementInNumericalSemigroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> GraphAssociatedToElementInNumericalSemigroup</code>( <var class="Arg">n, 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 {m_1,...,m_n} 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 { m_i | n-m_iin S}, and the second component is the set of edges of this graph, that is, { {m_i,m_j} | n-(m_i+m_j)in S}.</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.html#biBRos96">[Ros96a]</a>.</p>
<table class="example">
<tr><td><pre>
gap> s:=NumericalSemigroup(3,5,7);
<Numerical semigroup with 3 generators>
gap> GraphAssociatedToElementInNumericalSemigroup(10,s);
[ [ 3, 5, 7 ], [ [ 3, 7 ] ] ]
</pre></td></tr></table>
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