<?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> <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" /> </head> <body> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapC.html">C</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <div class="chlinkprevnexttop"> <a href="chap0.html">Top of Book</a> <a href="chap3.html">Previous Chapter</a> <a href="chap5.html">Next Chapter</a> </div> <p><a id="X7969F7F27AAF0BF1" name="X7969F7F27AAF0BF1"></a></p> <div class="ChapSects"><a href="chap4.html#X7969F7F27AAF0BF1">4 <span class="Heading"> Presentations of Numerical Semigroups </span></a> <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> </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. 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> <div class="chlinkprevnextbot"> <a href="chap0.html">Top of Book</a> <a href="chap3.html">Previous Chapter</a> <a href="chap5.html">Next Chapter</a> </div> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a href="chap1.html">1</a> <a href="chap2.html">2</a> <a href="chap3.html">3</a> <a href="chap4.html">4</a> <a href="chap5.html">5</a> <a href="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chapA.html">A</a> <a href="chapB.html">B</a> <a href="chapC.html">C</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div> <hr /> <p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p> </body> </html>