| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/sgpviz/doc/ (GAP Algebra Version 4.15.1©) Datei vom 30.7.2024 mit Größe 15 kB |
|
Quelle chap2.htmlSprache: 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> <title>GAP (SgpViz) - Chapter 2: Basics</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="chap2" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chap2_mj.html">[MathJax on]</a></p> <p><a id="X868F7BAB7AC2EEBC" name="X868F7BAB7AC2EEBC"></a></p> <div class="ChapSects"><a href="chap2.html#X868F7BAB7AC2EEBC">2 <span class="Heading">Basi <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7 </span> </div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F7FAED380682973">2 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83F1529479D56665">2 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7A65787A83C0F8EF">2 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7EEED52C7D38E1CA">2 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7BBEBEE885D05208">2 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7FB7633483A45209">2 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X822983CD7F01B5EA">2 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X82F7D3E485D615D8">2 </div></div> </div> <h3>2 <span class="Heading">Basics</span></h3> <p>We give some examples of semigroups to be used later. We also describe some basic functions tha <p><a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a></p> <h4>2.1 <span class="Heading">Examples </span></h4> <p>These are some examples of semigroups that will be used through this manual</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">f := FreeMonoid("a","b");</span> <free monoid on the generators [ a, b ]> <span class="GAPprompt">gap></span> <span class="GAPinput">a := GeneratorsOfMonoid( f )[ 1 ];;</s <span class="GAPprompt">gap></span> <span class="GAPinput">b := GeneratorsOfMonoid( f )[ 2 ];;</s <span class="GAPprompt">gap></span> <span class="GAPinput">r:=[[a^3,a^2],</span> <span class="GAPprompt">></span> <span class="GAPinput">[a^2*b,a^2],</span> <span class="GAPprompt">></span> <span class="GAPinput">[b*a^2,a^2],</span> <span class="GAPprompt">></span> <span class="GAPinput">[b^2,a^2],</span> <span class="GAPprompt">></span> <span class="GAPinput">[a*b*a,a],</span> <span class="GAPprompt">></span> <span class="GAPinput">[b*a*b,b] ];</span> [ [ a^3, a^2 ], [ a^2*b, a^2 ], [ b*a^2, a^2 ], [ b^2, a^2 ], [ a*b*a, a ], [ b*a*b, b ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">b21:= f/r;</span> <fp monoid on the generators [ a, b ]> </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">g0:=Transformation([4,1,2,4]);;< <span class="GAPprompt">gap></span> <span class="GAPinput">g1:=Transformation([1,3,4,4]);;< <span class="GAPprompt">gap></span> <span class="GAPinput">g2:=Transformation([2,4,3,4]);;< <span class="GAPprompt">gap></span> <span class="GAPinput">poi3:= Monoid(g0,g1,g2);</span> <monoid with 3 generators> </pre></div> <p><a id="X85134313846D1A8A" name="X85134313846D1A8A"></a></p> <h4>2.2 <span class="Heading">Some attributes</span></h4> <p>These functions are semigroup attributes that get stored once computed.</p> <p><a id="X7F7FAED380682973" name="X7F7FAED380682973"></a></p> <h5>2.2-1 HasCommutingIdempotents</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Tests whether the idempotents of the semigroup <var class="Arg">M </var>commute.</p> <p><a id="X83F1529479D56665" name="X83F1529479D56665"></a></p> <h5>2.2-2 IsInverseSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Tests whether a finite semigroup <var class="Arg">S </var>is inverse. It is well-known that it su <p><a id="X78CA2A0D869C51DC" name="X78CA2A0D869C51DC"></a></p> <h4>2.3 <span class="Heading">Some basic functions</span></h4> <p><a id="X7A65787A83C0F8EF" name="X7A65787A83C0F8EF"></a></p> <h5>2.3-1 PartialTransformation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>A partial transformation is a partial function of a set of integers of the form <span class="Sim <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">PartialTransformation([2,0,4,0] Transformation( [ 2, 5, 4, 5, 5 ] ) </pre></div> <p><a id="X7EEED52C7D38E1CA" name="X7EEED52C7D38E1CA"></a></p> <h5>2.3-2 ReduceNumberOfGenerators</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Given a subset <var class="Arg">L</var> of the generators of a semigroup, returns a list of genera <p><a id="X7BBEBEE885D05208" name="X7BBEBEE885D05208"></a></p> <h5>2.3-3 SemigroupFactorization</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p><var class="Arg">L</var> is an element (or list of elements) of the semigroup <var class="Arg">S</v <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">el1 := Transformation( [ 2, 3, 4, 4 ] ); <span class="GAPprompt">gap></span> <span class="GAPinput">el2 := Transformation( [ 2, 4, 3, 4 ] ); <span class="GAPprompt">gap></span> <span class="GAPinput">f1 := SemigroupFactorization(poi3 [ [ Transformation( [ 1, 3, 4, 4 ] ), Transformation( [ 2, 4, 3, 4 ] ) ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">f1[1][1] * f1[1][2] = el1;</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">SemigroupFactorization(poi3,[el [ [ Transformation( [ 1, 3, 4, 4 ] ), Transformation( [ 2, 4, 3, 4 ] ) ], [ Transformation( [ 2, 4, 3, 4 ] ) ] ] </pre></div> <p><a id="X7FB7633483A45209" name="X7FB7633483A45209"></a></p> <h5>2.3-4 GrahamBlocks</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p><var class="Arg">mat</var> is a matrix as displayed by <code class="code">DisplayEggBoxOfDClas <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">p1 := PartialTransformation([6,2, <span class="GAPprompt">gap></span> <span class="GAPinput">p2 := PartialTransformation([0,0, <span class="GAPprompt">gap></span> <span class="GAPinput">p3 := PartialTransformation([0,0, <span class="GAPprompt">gap></span> <span class="GAPinput">p4 := PartialTransformation([4,4, <span class="GAPprompt">gap></span> <span class="GAPinput">css3:=Semigroup(p1,p2,p3,p4);</s <transformation semigroup of degree 13 with 4 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">el := Elements(css3)[8];;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">D := GreensDClassOfElement(css3, e <span class="GAPprompt">gap></span> <span class="GAPinput">IsRegularDClass(D);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">DisplayEggBoxOfDClass(D);</span> [ [ 1, 1, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">mat := [ [ 1, 0, 1, 0 ],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 1 ],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [ 0, 1, 0, 1 ],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [ 1, 0, 1, 0 ] ];;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">res := GrahamBlocks(mat);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">PrintArray(res[1]);</span> [ [ 1, 1, 0, 0 ], [ 1, 1, 0, 0 ], [ 0, 0, 1, 1 ], [ 0, 0, 1, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">PrintArray(res[2]);</span> [ [ [ 1, 1 ], [ 1, 3 ], [ 1, 2 ], [ 1, 4 ] ], [ [ 4, 1 ], [ 4, 3 ], [ 4, 2 ], [ 4, 4 ] ], [ [ 2, 1 ], [ 2, 3 ], [ 2, 2 ], [ 2, 4 ] ], [ [ 3, 1 ], [ 3, 3 ], [ 3, 2 ], [ 3, 4 ] ] ] </pre></div> <p><a id="X789D5E5A8558AA07" name="X789D5E5A8558AA07"></a></p> <h4>2.4 <span class="Heading">Cayley graphs</span></h4> <p><a id="X822983CD7F01B5EA" name="X822983CD7F01B5EA"></a></p> <h5>2.4-1 RightCayleyGraphAsAutomaton</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Computes the right Cayley graph of a finite monoid or semigroup <var class="Arg">S</var>. It uses <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">rcg := RightCayleyGraphAsAutomato < deterministic automaton on 2 letters with 6 states > <span class="GAPprompt">gap></span> <span class="GAPinput">Display(rcg);</span> | 1 2 3 4 5 6 ----------------------- a | 2 4 6 4 2 4 b | 3 5 4 4 4 3 Initial state: [ ] Accepting state: [ ] </pre></div> <p><a id="X82F7D3E485D615D8" name="X82F7D3E485D615D8"></a></p> <h5>2.4-2 RightCayleyGraphMonoidAsAutomaton</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>This function is a synonym of <code class="func">RightCayleyGraphAsAutomaton</code> (<a href= <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html>
¤ Dauer der Verarbeitung: 0.16 Sekunden (vorverarbeitet am 2026-06-23) ¤*© Formatika GbR, Deutschland |
|
||||||||||||||||
2026-07-27