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<?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.3.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_HTMLor </script> <title>GAP (NumericalSgps) - Chapter 3: Basic operations with 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="chap3" onload="jscontent()"> <html xmlns"http://www.w3.org/1999/xhtml" xml:lang="en"> <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a java.lang.StringI <<inkrel=stylesheet=""ttp://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <p><a id="X7A9D13C778697F6C" name="script src="manual.js" type="text/javascriptjava.lan <=C">href="hap3_mjhtmlX7A9D13C778697F6Cjava.lang.StringIndexOutOfBoundsException: java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 0 /><a <div class="ContSect" Invariants </span></a> </ operations with numerical semigroups <div class="ontSSBlock"> <span class="ContSS"><br /><span</title> <panclass="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#java.lang.StringI <span </span></a> <spanContSS"br/><span class="nocss"> </span><a href="chap3_mj.html#X84A6B16E8113167B">3.1-4 spanclass="ocss>&bsp&bsp;<span><a href=chap3_mj.html#X7A56569F853DADED">3.1-5 Length</a></span> <span class="ContSS"><br /><span </span</> <span java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length < /><spanclass="ocss"> </span><a href="chap3_mj.html#X81A2505E8120F4D7"><code>3.1-8 <span>\</span> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7A34F16F8112C2B <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X84345D5E7CAA9B7 <pan =ContSS"<r /span class="nocss"> </span><a href="chap3_mj.html#X7B6C82DD86E5422F">3.1-11 <java.lang.StringIndexOutOfBoundsException: Range [38, 5) out of bounds for lengt <span class=ContSS"<r /><pan class=nocss"> </span><a href="chap3_mj.html#X867ABF7C7991ED7C">3. </><spanclass"nocss" /span><a href="chap3_mj.html#X7E6F5D6F7B0C9635">3.1-14 Differen <<spanclass="ContSStt"><span class=tocline"><span class="nocss"> </span><a href="chap3_mj.html <span class="ContSS"><br /><span class="nocss"> </<pan class="ContSS"><br /><span class="nocss"> </spa <class"ContSS><br /<span class="nocss<pan class="ContSS"><br /><span class="nocss"> </span><a href=" java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length spanC"br >span class="ocss>nbsp&<span> href"chap3_mj.tml#X80B398537887FD87>3119KunzCoo <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7C21E5417A3894E <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7802096584D3279 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X847BAD9480D186C ="ocss"&&;/pan<href=chap3_mjtml729D7D8B1B36java.lang.StringIndexOutOfBoundsExceptio <span class="ContSS"><br /><span class="nocssspan ="ontSS>br/<span =""&;;span> =c.html#E6F5D6F7B <="ContSS"</>spanclass="nocss">&bsp;&bsp;/span>ahref=chap3_mjhtml#865"3.1-5Type<a></> <><a href="chap3_mj.htmlX8688B1837E4BC079"3.-26Gaps/></spanjava.lang.StringIndexOutOfB <span class="ContSS"><br /><span classspan class="ContSS"><br /><span class=<span =ContSS"< /<class= <span class="ContSS"><br /><span class="nocss"> </span>span> spanclass=ContSS"< >span=nocss"> </span><a href="chap3_mj.html#X82B1868F7A780B49">3.1-29 Is <span">3.1-30 IsAcute</a</pan> <span class="ContSS"><br<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.h <span class</div>/iv> <span class <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7EC438CC7BF539D <<span class="nocss"> </span><a href="chap3_mj.html#X803D550C78717A7C">3.1-35 SpecialGaps</a< </div><div> <<span class=ContSS"><br />span =""&;&bsp;<>a ="chap3_mjhtml">3.124 </>/spanjava.lang.StringIndexOutOf </span><>31class""> <divclass="ontSSBlock"> <span class="ContSS"><br /><span java.lang.StringIndexOutOfBoundsException: Range [0, 38 <java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length <=ContSS"<r/>< class=nocss"n;n<< href"html# <java.lang.StringIndexOutOfBoundsException: Range [19, 5) out of bounds for lengt <>/div </div> <h3>3 <span class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds f div class="func"> =" ="00%<<tdclass"java.lang.StringIndexOutOfBoundsException: Range [ /span>h3> <p describessome java.lang.StringIndexOutOfBoundsException: Range [68, 46) out of b <p><a id="X87AF9D4F7FD9E820span ="nocss"&; </< href="chap3_mj.html#X7CCFC5267FD27DDE">3.1-31 <h4>3.1 <span class="Heading"> Invariants <><4java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 28 <p>In this java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for leng <p><a idspan class="Modularnumerical semigroupsatisfying mod 53 <=x g; </s>/><div> <iv class="func"><table class="func" width=span class="ontSS"<br class=java.lang.StringIn <iv =func"< class=func ="100%"><tr>td classjava.lang.StringIndexOutOfBoundsException: Ind <p><><varclass=A"NS</var> is <span ="">br/>span class">nbsp;;/span>a href".#X7B45623E7D539CB6>323P <div class="example"><pre> <divclass"func"><table class="unc width="100%"><tr><td class="tdleft"><code class="func">‣ Minim cal with 2 generatorsgt; <span Basic operationswith numericalsemigroups 3 <span class="GAPprompt">gap>&t;odular numerical semigroup satisfying 5x mod 53 <= x > &tModular numerical semigroup satisfying 7x mod 53 <= x > <pan class="APprompt">gap></span> <span class="GAPinput">MultiplicityOfNumericalSemigroup( 8 </pre></div> <p><a id="X850F430A8284DF9A" name="X850F430A8284DF9A"></a></p> <h5>3.1-2 Generators</h5> <div class="func"><table class[3 5,53 ] < class="unc width="100%"><r>tdclass"dleft>code class="func>&8227;GeneratorsOfNumericalSemigrou <div class="span ="GAPprompt"gap&;</pan> <panclass="GAPinput">MinimalGeneratingSystemOfNum <div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Index 36 o <div class="unc"<tableclass="func" width="100%"><tr><td class="tdleft"><code class="func">&# <=Arg>S</var> is a semigroup.code "code <div class="example"java.lang.StringIndexOutOfBoundsException: Index 5 out of boun <span class=="Pprompt">gap&;<span> <span class="GAPinput">S := NumericalSemigroup("modular", 5 <Modularscript type"text/javascript">overwriteStyle();</script> <java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for lengt 11,12,13,32 53 java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds f <="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup(3, 5, 53);</span <p><a idX7884AE27790E687F" name="java.lang.StringIndexOutOfBoundsException: Index 37 <span class="GAPprompt<h5>3.-3 EmbeddingDimension</h5> [,5,53 ] <span pdivclass="unc">table =func width=100"tr> class=""><code class=div class="unc"class=f [3,5 ] <span class= Invariants [3,< class="APprompt">t<span spanclassGAPinput>(s;/java.lang.StringIndexOutOfBoundsExc <">gap><span>spanclass""()MinimalGeneratingSystemOfNumericalSemigroup()</java.lang.Str true <java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for lengt &java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <span class true <span java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length false <span class="GAPprompt">gap<pan ="ContSS"><br />< [pan <divclass"java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bo <span class=GAPprompt , 5, 7,15 </pre></div> <p><a id="<pan ="ContSS"><br /><<p><code class="code">NS</code> a numerical semigroup. It returns the lis <h5>3.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length <div class="func"<table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <ivclass="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le <p>span class"<br /<span class==nocss">nbsp<;&bsp;span><a hrefchap3_mjh#X81A2505E8120F4D7"<which is <div class="example"><p with2generators&; <pan class="GAPprompt">gap></span> <span class="GAPinput">s: &th 4 generators> < class="GAPprompt">span class="GAPprompt">gap;</span><span class="GAPinput">Length(Numeri 3 <span class="GAPprompt">gap></span> <span class="GAPinput">EmbeddingDimensionOfNumericalSe 3 </pre></div> <p><a id="X84A6B16E8113167B" name="X84A6B16E8113167B"></ <h5>3.1-4 SmallElements</<p>code <span java.lang.StringIndexOutOfBoundsException: Range [ <div class="func">< <div class=" <divclass"xample"><pre> <div class="example"><pre> <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for l ,3 ] <span class="GAPprompt">gap></span> <span span class="GAPprompt">gap></span> <span class(10,Numer 0,5 </pre></div> <><a id="756569F853DADED" name="X7A56569F853DADED"></a></p> <h5>3.1-5 Length</h5> <"uncp><r =Ag"S/varnumerical semigroup.<codeclass"ode>GeneratorsOfNumericalSemigroup</co <p<code class=="ode>NS</code> is a numerical semigroup. It returns the number of small elements of < <div class <span class="GAPprompt"><div class="xample"><pre> 2 </preclass="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfNumericalSemigroup( <p><a id="X7F0EDFA77F929120" name="X7F0EDFA77F929120">spanclass="[0, 5, 7, 10, 12, 14, 15, 17 <h5>3.1-6 FirstElementsOfNumericalSemigroup ] <div class="func"><table class="func" spanclass=GAPprompt">gap></span> <span class="GAPinput"> <odeclass="ode>NS</code> is a numerical semigroup. It returns the list with the first <var class=" <iv class="xample"><pre> <span class=GAPprompt">gap></span> [ 0, 3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBound [ 0 ,[3 5 7 15 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for l /pre>/divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for le > id=XD2B3AA9823371AE" name="X7D2B3AA9823371AE"></a></p> <h5>3.1-<<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup(7,8 <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func68 <pvar ="rg">S/var is ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bound <div class="example"><pre> <class="span class="GAPprompt>p&;</pan <span class="GAPinput">s := NumericalSemigroup(3,5,7, <ss="APprompt">gap></span> <span class="GAPinput"&tNumerical semigroup with 4 generators> 0, 5,,10,,4 15,17,19 20,1,22,24 ] < class=GAPpromptggt;/> <span class=GAPinput>(ns18java.lang.StringIndexOutOfBoundsExcep [ , 12, 14,, 15, 17 ] <<span 5,7 , ,15,17 19 ,21 22 24 25, 26,27 ] </re><div <p><a id=" <code31-8<span>\</span>[ <span>\</span>]</code></h5> <div java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 psemigroupand< =Arg></ aninteger It returns <varclass=Arg"r<var>th of <var class=ArgS<var><p> <class=example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">S := span class="GAPprompt">gap></spa <span class="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for 68 </pre></div> <p><a id="java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for leng <h5><code>3.1-9 DD86E5422F" name="X7B6C82DD86E5422F"></a></p> <div <<var class"">/>isnumerical and<var ="Arg"></> is a listof<div RthElementOfNumericalSemigroup/co <div class="example"><pre> <java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for lengt span class="GAPprompt">gap></span> <pan class="GAPinput">S{[1..5]};</span> [0,7,8,1415 ] <//re><div> <p <h5>3.1-10 NextElementOfNumericalSemigroup</java.lang.StringIndexOutOfBoundsExceptio <div class="func"><table class="func" width="100% <p>var ="Arg><var is a numerical semigroup numerical semigroup andvar class="Arg"r/var> is an int <div class="example"><pre> <span class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for <span class=<p><a id"X7F0EDFA77F929120" name><ajava.lang.StringIndexOutOfBoundsException: 14 <span class="<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigrou 17 <<h5>3.-6 FirstElementsOfNumericalSemigroup</5> true </pre></div> <p><a id="X7B6C82DD86E5422Fp<code class="code> <h5>3.1-</h5> <">< class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementNumber_Numeric <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p><var class="0 <div class="example"><pre> <span class="GAPpromptspan class="GAPprompt">gap></span> <span class="GAPinput">NextIterato <span classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for le 68 <span class="GAPprompt">gap></span> <span class="GAPinput">RthElementOfNumericalSemigroup( 68 </pre<iv ="unc"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#style='color: green'>8227; Difference</code>( <div class=p<arclass="rg>NS</var> is numerical s <p><a id="X78F4A7A7797E26D4" name="spanclass="GAPprompt">gap&[ 0, 5, 7, 10, 12, 14, 15, 17, <h5>3.1-12 java.lang.StringIndexOutOfBoundsException: Index 23 out of bounds for le <ivclass="func"<tableclass=func width="00"<>class="> class=f">#8227; java.lang.StringInde <java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length <<div class"unc">table class="func" width="100%"><tr<pan class=ContSS"< /< ="ocss"&;&bsp<>< class""S</> <span class="GAPprompt">gap></span> <span class="GAPinput"></re/div> < ="GAPprompt">gap;/> <span class=GAPinput>java.lang.StringIndexOutOfBoundsException: In 53 </pre></div> <p><a id="X867ABF7C7991ED7C" name="X867ABF7C7991ED7C"></ajava.lang.StringIndexOutOfBoun <h5>3.1-13 Iterator</h5> <div class="func"><table class="func" width="100%"><tr><td class="GAPprompt">gap></span> <span cla <<var class="g">/> is a Itreturns aniteratorover < =Arg></./> <div class="example"><pre> <span class="GAPprompt">gap&<pan class="GAPprompt">gap></span> <span class="GAPinput">First <span class="GAPprompt">gap></span> <span <iv ="unc"<table <>apgt;<span><span class="APinput">S := Num <iteratorgt; <span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator(iter);</span> 0 <span class="GAPprompt">gap></span> <span<<<ar class=r"S/var> is a numerical semigroup. It computes 7 <java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for lengt 8 </<p><v class="Arg>S</ar>is a numerical class=Arg"ls</var> is a list of integers.span class="GAPpro <p << <div class="func"><table [ ,8 14, 15] < class"func"<table/pre>/java.lang.StringIndexOutOfBoundsException: Index 12 out of <p><var class="Arg">S, T</var<iv ="unc"<table class="func" width="100%"><java.lang.StringIndex <div class="div ar class="Arg"S</ar>is numerical semigroup and < =Arg"><var is an . Itspanclass"">g <pan="">g;</><span class""ns1:=NumericalSemigroup(57;;/java.lang.StringIndexOutOfBound <pan class="GAPprompt">gap></span> <span class="GAPinput">ns2 := NumericalSemigroup(7,11,12) <pan class=GAPprompt">gap></span> <span class="GAPinput">Difference(ns1,ns2);</span> [ 5, 10, 15, 17, 20, 27 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Difference(ns2,ns1);</span> [ 11, 18, 23 ] <span class="GAPprompt">gap></span> <span class="GAPinput">DifferenceOfNumericalSemigroups [ 11,18,23 ] </pre></div> 93E1 name="X7CB24F5E84793BE1"</></java.lang.StringIndexOutOfBoundsException: Index 6 <h5span class="APprompt">gap></span> <span class="GAPinput">AperyListOfNumericalSemigroupW <div class="func"><table class="func" width="100%"><tr><td class="tdleft">panclass="APprompt <div [ 0 51 32,13,54 45 <>varclass=Arg>S</var> is class=Arg>n/var a elementofvar10 <div class="example"><pre> <span class="APprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup("modular", <span class="GAPprompt">gap></span> <span class="GAPinput">AperyList(S,12);</span> [,<varclass=Arg"</ar a numericalsemigroupand <r ></ an.Both functions ( are synonyms)returnthe class=GAPprompt>gap&<span>span =GAPinput>peryListOfNumericalSemigroupWRTElement10)<s [ 39,52,3,54 ,32,>span class=APinput" (,,17;/java.lang.StringIndexOutOfBoundsExceptio <panclass"APprompt">gap></span> <span class="GAPinput">First(S,x-> x mod 12 =1);</span> 13 </pre></div> <p><a id="X80431F487C71D67B" name="<a id="X80431F487C71D67B" name="X80431F487C71D67B>/p> <h5>3.1-16 AperyList</h5> <div class="func">span class"">br/<class=java.lang.StringIndexOutOfBoundsException: Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p< class=ContSS>br/<class="">&; << =c.tmlX847BAD9480D186C03.- java.lang.StringIndexOutOfB <div class="example"><pre> <span class="GAPprompt">gap></<span class="ContSS"><br /><span clasnbsp&bsp>ahref"hap3_mj.html# 14 ] <panclass=GAPprompt"&t;/span>< =java.lang.StringIndexOutOfBoundsException: Range [61, 6 <span class=GAPprompt"gap&</pan>< =GAPinput"AperyListOfNumericalSemigroup> [0,12,13 25,26 38,39 51, 52 53,32 java.lang.StringIndexOutOfBoundsException: Index 45 <div="xample"<> p>a id"&;Numericalsemigroup 3 generatorsg; <pan ""gapgt;/span <pan class">KunzCoordinates(s;<span> <h5>3.1-17 AperyList</h5> <java.lang.StringIndexOutOfBoundsException: Range [99, 4) out of bounds for lengt <div class="func"><table <pan class="GAPprompt">gap></span> <span class="GAPinput">KunzCoordina <rical semigroup and <var class="">mvar is integer.Computesthe Apérylist <var class=Arg"></var wi <div class="example"><pre> <span class="GAPprompt">gap><<p>var class="Arg">m</var> is a positive integer.</p> <span class=div>/div> [ 0, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <span class="GAPprompt">gap></span> <span class="GAPinput"<span class="GAPprompt"">gap><span <sp [ 0, 10, 13, 19, 20, 23, 26, 27, 29, <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 18 <span class="GAPprompt">gap></span> <span class [ 0, 13, 19, 26, 27, [div class="unc"<table class="func" java.lang.StringIndexOutOfBoundsE <promptgap&;<spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds fo [ 0, 51,,32, 13,54 26>s numericalsemigroup Returnsthe multiplicity class"">NS</>,which the sma "GAPprompt>gap&t</span> <pan class""s10);/pan> [ <span ="GAPprompt">><span><=""Length)< 10 </div"unc< " =100%>tr class"dleft"<>8227; <>(var=><<Thejava.lang.StringIndexOutOfBoundsExcepti <p><a id="X8022CC477E9BF678" name="X8022CC477E9BF678"></a></p> <h53118 peryListOfNumericalSemigroupAsGraph</h5> <div<div span classGAPpromptgap>/pan><class"APinput">(NumericalSemigroupivclass=fnc>class <p><var class="Arg">ap</var> java.lang.StringIndexOutOfBoundsException: Range [0, 31) ou java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <> <span class=Gjava.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for <divclass="xample"<> , 21 ,3 14, 15, 6< classfunc" "uncwidth=%>tr<td class=tdleft="=%">trjava.lang.StringIndexOut <span class="GAPprompt">gap></span> <span class="GAPinput">AperyListOfNumericalSemigroupAs [ ,, [ 3, 6, 9, 12, 15, 18, 21 ],,, [ 6, 9, 12, 15, 18, 21 [ 7,14 21 ],,[ < class=GAPprompt">ap&;/>< class="GAPinput> : NumericalSemigroup(7817);<span> [ 14, 21 ], [ 15java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for /re <p><ah5.1-3Iteratorh5 <h5>119KunzCoordinates> <class""< class" idth"100"<tr><td class="tdleft"><code class="func">‣ <p>An integer <span class="Sim <div class="func"><table class="func" width="100%"<span class="APprompt">gap></span> <span clas <p><var class="<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigrou <p>Then the Apé set of ="rg"m/ar inar =Arg"S<var> has form <span class="SimpleMath">\([0,k_1m+1,.. <div java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 1 <span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 <Numerical semigroup with 3 class=G">&;<span><span class="GAPinput">KunzCoordinates(s);</span> [ 2, 1 ] <pan class">.25Type/5> [ 2, 1 ] <pan class>&t;/> span ="APinput">unzCoordinatess5);/< ""<table "100"<tr< ""< class"unc">‣ TypeOfNumericalSemigroup</code>(p>Standsfor < class="ode">ength(seudo [ 1, 1, 0, 1 ] <span class=GAPprompt"gap><span> <span class="GAPinput">p><var class="Arg">S, T</var> are numerica [,1 0,] </prespan class"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of b <p><a id="X7C21E5417A3894EC<span class="GAPprompt">gap></span> <span class="GAPinput">Differen <h5>3.1-20 KunzPolytope</h5> <div class="func"><table class="func" width="100%"><tr><td class[11, 18,23] <><11,1823] <The coordinates< <java.lang.StringIndexOutOfBoundsException: Range [120, 4) out of bounds for leng <span class="GAPprompt">gap></span> <span class="GAPinput">KunzPolytope class="unc">< class"un [ [ 1, 0, -1 ], [ 0, 1, -1<h5< ="GAPprompt>/> <span class="GAPinput"S : NumericalSemigroup(" 5,);/sp </pre></div> <p><a 0 ,26,39, 52 53, 54 43,32, 33 22,11] <java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length <div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 1 out of bo <p><var class="Arg">S</var> is a numerical semigroup <div class="example/re></div> <span">gap&t;/span> <span class=lt;Modularnumerical semigroup<p" name="X80431F487C71D67B"></a></ <span<div class="func"><table class="func" width="100%" [ 0,0 0, 3 4 ],[0,4,1 ] ] <>""able= =100"tr< =dleft>code java.lang.StringIndexOutOfBoundsException: Range [86, 85 <p>a =">ar class="rg>/ isa semigroup.Itcomputesthe Apéry java.lang.StringIndexOutOfBoundsEx <h5>3.1-22 FrobeniusNumber</h5> <div=<pan ="Pprompt"gapgt<spanjava.lang.StringIndexOutOfBoundsException: Index 39 ou <iv class=func>tableclass="width"100%><r<tdclass=tdleft>< =func"‣ <r notjava.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for <p><code class="code">NS</code>h5>31-9IsOrdinary</h5> < =example<ivclass""< class""width=100%"tr>td=tdleft"><ode class="unc"&8227 IsOrdinaryNumericalSemigroup<code>( << =rg><var a . Dectects semigroupis ordinary , l <span class="GAPprompt">gap 4 <span class="GAPprompt">gap></span> <span class=,5,7 15] 4 </pre></ <p><a id="X835C729D7D8B1B36" namespan class="GAPprompt">gap></span> <span java.lang.StringInd <h5>3.1-23 Conductor</h5> <div div"nc><able ="unc =100%>tr> class"dleft"< =func">&8227;</><var ="Arg"NS/var> )/td>td ="&attribute< <p>This is just a synonym <div class="example"><pre> <panclass=GAPprompt&;</> span class=GAPinputConductorN(357)<span> 5 < =GAPprompt>gt</pan <pan =GAPinput>(NumericalSemigroup35,);/> 5 </pre></divjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for le <p><a id="X861DED207A2B5419" name="X861DED207A2B5419"></a></p>3 <h5>3.1-24 PseudoFrobenius</java.lang.StringIndexOutOfBoundsException: Index 4 out o <div class="func"><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds <div class="func"><table class=< class="unc"tableclass"width=100 <p>An <>class""S/>is numericalsemigroup returns the<iv "unc">table class=func width"100%>>td class=t <div class="example"><pre> <span class="GAPprompt">ap>/span> <<an class=GAPprompt>gap>span span ="((3,57)</pan> &;odular semigroupsatisfying 5 mod &; &; <span class="APprompt">gap></span> [ 21, 40, <span class="GAPprompt">gap></span</pre></div> [ <<a="A56569F853DADED" name="X7A56569F853DADED"></a></p> </pre></div <p><a <div cla class"unc"<table ="unc" width"100%><tr>tdclass"tdleft"<code class="func">‣ Length</c <h5>3.1-25 Type</h5> < class"">table =func" =100java.lang.StringIndexOutOfBoundsException: Index 0 out of < =funcrompt"&t<span><span ="APinput>=NumericalSemigroup16,,,72);;</span> <p>tands for< <class"example"><re> <[,3] <Modular numerical class=GAPprompt>><span <spanclass=java.lang.StringIndexOutOfBounds <class"APprompt"gap&;/><panclass=GAPinput>(;/pan> 4 <span p<id"7D2B3AA9823371AE" name="X7D2B3AA9823371AE"> 4 </pre></div> =div class="func">class"" =100"<>td java.lang.StringIndexOutOfBoundsException: Range [ <h5>3.1-26 Gaps</h5> <ivclass""> <"unc"width="00>tr>td class=java.lang.StringIndexOutOfBoundsException: Range [74, 72 <>A<emgap<em of a numerical</span> <span class="APinput">s:=NumericalSemigroup(16,17,71,72);;< <div class="example"<panclass=GAPprompt"gap></span>< java.lang.StringIndexOutOfBounds <Pprompt>gt;</80 [ 1, 2, 3, <pan class="GAPprompt">gap></span><pan =GAPprompt"gapgt;/pan><span classclass=GAPinp <span class="GAPprompt">gap></span> <span class="GAPinput">GapsOfNumericalSemigroup(Numeri [ 1, 2, 4 ] </pre></div>h5<>.- span><>\/> <p><a id="X7F71983880DF4B9D" name=divclass=func <h5>3.1-27 Weight</h5> < =func>table =func"width=100"<>java.lang.StringIndexOutOfBoundsException: Range [0, 5 <java.lang.StringIndexOutOfBoundsException: Range [22, 2) out of bounds for lengt <div class="example"><pre> <span class=GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Range [0, 30) out of bounds for length 0 0 <class=GAPprompt">>/><span class="GAPinput">Weight(NumericalSemigroup(4,5)); </span> 9 </pre></div> <><a id"X7EB81BF886DDA29A" name="X7EB81BF886DDA29A"></a></p> <5>3.28Deserts</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div ="">table class=func"width=100%>tr>td =""class""‣ </code(<var class=Arg"NS/>)>td =" 30 31 0,4 <p><var class="Arg"> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</ <span class="GAPprompt">gap></span> <<"name=java.lang.StringIndexOutOfBoundsException: Ra [ [ 1, 2 ], [ 4 ] ] <panclass="">ap&t</>< class=GAPinput"DesertsOfNumericalSemigroup);/> java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 19 <<> <m>special gaps</em of numerical semigroup <span class="SimpleMath">\(S\)</span> are those funda >=X82B1868F7A780B49=X82B1868F7A780B49/><> h5.<h5> <div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Index 34 o <div class="func"><table class="func" width="100%"><tr>< <java.lang.StringIndexOutOfBoundsException: Range [92, 2) out of bounds for lengt <pThisfilter impliescodeclass="unc"IsAcuteNumericalSemigroup</code> (<a href="chap3_mj. <divclass="example"><pre> <s=GAPprompt>;span <span =">:( < =APprompt>gt<java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of boun false </pre></div> <>a =XDA"=""<a><< <h5>3.1-30 java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for le <iv class"unc<able class="unc"width="100%"><tr><td class="tdleft"><code class=java.lang.Stri <div class="func"><table""width"%>tr> class="< =>#;IsAcuteNumericalSemigroup>< =A"/>)/<class <p><var class="Arg">NS</var> is a numerical semigroup. Dectects if the semigroup is <span class="GA <div class <span <class=<span class="APprompt">ap&t</span><span class="GAPinput">Maximum(Set(l, s->WilfNumb true </pre>div> a></p> <h5>3.1-31 Holes</h5> <div class="func"><table class="func" width="100%"><tr><iv class=func>table class="func" width <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class=" <p><code class="code">S</code> is a java.lang.StringIndexOutOfBoundsException: Index 37 ou <div class="example"><pre> java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length <spanclass="GAPprompt">gap></span> <span class="GAPinput">Holes(s);</span> [ ] <span class="GAPprompt">gap></span> <span class="><aid="" name=java.lang.StringIndexOutOfBou <java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 0 G"gapgt<span>span class=G>(s1,;<> </pre></div> <p><a id= <h5>3.1-32 java.lang.StringIndexOutOfBoundsException: Index 34 out of bounds for le <classfunc>tableclass" =100%>tr><td class="tdleft"><code class="func">‣ LatticePathAssociated <p><531-java.lang.StringIndexOutOfBoundsException: Range [22, 20) out of bounds for <p>In this setting <code class="code"<h5>3.-4 EliahouSlicesOfNumericalSemigroup</h5> <The is < class"func"<class"unc =100%>tr>td="< ""&8227;EliahouSlic < class"GAPprompt">gap;/span<class"APinputAperyListOfNumericalSemigroupWRTElement12) <panclass""gapg;/> < class"GAPinput>:NumericalSemigroup16,177172);/> <span class="GAPprompt">gap></span> <span class="GAPinput">LatticePathAssociatedToNumerica [ [ 0, 14 ],<pre</iv> [7 7],[ 8 6 ] [,,[10 4],[ 11, ],[ 12,2], [ 13 1 ], [14, ] <<div <p><a id="X7E9C8E157C4EAAB0 <h5>3.1-33 Genus</h5> <ivclass""< java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for le <div class="func"<able =spanclass="gapgt<span < =GAPinput"AperyListOfNumericalSemigroup( <p><code class="code">NS</code> is/<div><hr/ <div class="example"><pre> <pan class=GAPpromptgt<>spanclass="">=(16177172)<span> <n> spanclass"java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of boun 80 <span class="GAPprompt">gap></span> <span class="GAPinput">GenusOfNumericalSemigroup(s);</s 80 <span class="GAPprompt">gap></span> <span classspan class="GAPprompt">gap></span> <span class="GA <Modular numerical semigroup satisfying 5x mod 53 <= x > <panclass="GAPprompt"gap>/span <panclass=GAPinput"Genus(S);</span> 26 <pre<div> <p><a id="java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for leng java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 31 <ass">code class=>&8227;</code>(<var =A"< <td=tdright>&;ttributenbsp)/d><tr/>/> <div class="func"><table class="func" width="100%"><tr><td class="java.lang.StringIndexOutO <p><code class="code",51 32 ,54,,26 ,38 19 java.lang.StringIndexOutOfBoundsException: Ind <div class="example"pa=X8022CC477E9BF678""></</> <<>.118AperyListOfNumericalSemigroupAsGraph [ ,8,9 13 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for len <=GAPpromptgapgt<span span =GAPinput> =NumericalSemigroup" ,)< <Modular numerical semigroup satisfying 5x mod 53 <= x > <span class="GAPprompt">gap></span> <span class="GAPinput">FundamentalGapsOfNumericalSemig [ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42 ] <span class="div class="example"><pre> [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29, 30 31 ,41,42 ] <span class=[ 0,21,12,3,14,15,,7,18 9] [ 1, 2, 3, 4, 6, 8, 9, 13 ] </pre></div> <><aid="X803D550C78717A7C" name="X803D550C78717A7C"></a></p> <h5>3.1-35 SpecialGaps</h5> <iv class="unc"<table class=func"width"100%><r<td =tdleft"><code class="func">‣ SpecialGaps</co <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>The <emdiv class="func<table class="func" width="100%"><tr><td class="tdleft"><code class="fu <div class="example"><pre> <span class="GAPprompt">gap<Numerical semigroup 3 generators> <Modular numerical semigroup satisfying 5x mod 53 <= x > <span class="GAPprompt">gap></span> <span class="GAPinput">SpecialGaps(S);</span> [ 40, 41, 42 ] <span class="GAPprompt">gap></span> <span class="GAPinput">SpecialGapsOfNumericalSemigroup [ 40, 41, 42 ] </pre></div> <p><a id="java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for leng <h4>3.2 <span class="Heading">Wilf's conjecture</span></h4> <p>Let <span class="SimpleMath">\(S\)</span> be a numerical semigroup, with conductor <span class <>a id"X78C2F4C77FB096F0" name="X78C2F4C77FB096F0"></a></p> <h5>3.2-1 WilfNumber</h5> <div class="func"><<p>The Kunz coordinatesthesemigroups with multiplicity <var class="Arg">m</v <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p><code class="code">S<java.lang.StringIndexOutOfBoundsException: Index 0 out of boun <iv ="example"><pre <<<var class="Arg"S/>is a numerical semigroup, and <var class="Arg">m</var> is a nonzero element of <v <span class="GAPprompt<pan class="GAPprompt">gap></span> <span class="GAPinput">CocycleOfNum 96 <span class="GAPprompt">gap></span> <span class="GAPinput">l:=NumericalSemigroupsWithGenus ">&/pan>< class="Filteredl ->()<0;/span [ <span class"GAPprompt"gapgt<span> <span class="GAPinput">Maximum(Set(l, s->WilfNumberOfNum 70 </pre></div> <p><a id="X80F9EC9A7BF4E606p> not a semigroup <span class=Sjava.lang.StringIndexOutOfBounds <h5>3.2-2 EliahouNumber</h5> <class"unc"<ableclass="func" width="100%"><tr><td class="tdleft"><code class="func">&#le='color: green'>8227; EliahouNumber</code>( <var class="Arg">S</var> )</td><td java.lang.Stri <div class4 <p><code class="code">S</ <div class="example"><pre> <java.lang.StringIndexOutOfBoundsException: Range [26, 5) out of bounds for lengt <pan class="GAPprompt">gap></span> <span class="GAPinput">EliahouNumber(s);</span> -1 <span;</span> spanclass="GAPinput">s:=NumericalSemigroup(5,7,9);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">TruncatedWilfNumberOfNumericalS 4 </pre></div> <p><a id="X7B45623E7D539CB6" name="X7B45623E7D539CB6"></a></p> <h5>3.2-3 ProfileOfNumericalSemigroup</h5> <class="func><able class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Profile <p><code class="<p> is a synonym of <ode <iv"example">pre <span class="GAPprompt">gap>5 <span class="GAPprompt">gap></span> <span class="GAPinput">ProfileOfNumericalSemigroup(s);< [ 2, 1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroupWithGivenE <span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 37 out [ 3, 0, 0 ] </pre></div> <p><a id="X7846F90E7EA43C47" name="X7846F90E7EA43C47"></a></p> <h5>3.2-4 EliahouSlicesOfNumericalSemigroup</h5> <div java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0 <> class""Sis numerical <spanclass"impleMath">\c\)/pan span class="SimpleMath">\(m\)</span> b <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,9);; </ <span class="GAPprompt">gap></span> <span class="GAPinput">EliahouSlicesOfNumericalSemigro [ [ 5, 7 ], [ 9, 10, 12 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput>< id =func"<java.lang.StringIndexOut ,5 7, ,9 10 12 14 ] </pre></div> < Book]/&;< java.lang.StringIndexOutOfBoundsException: Range [94, 93) out of bounds fo <<div""&; "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds fo < / <pated < =":/math..e/./GAPDoc></></p> <java.lang.StringIndexOutOfBoundsException: Range [7, 6) out of bounds for length <
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