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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.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (Semigroups) - Chapter 3: Bipartitions and blocks </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()"> <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="chap3_mj.html">[MathJax on]</a></p> <p><a id="X7C18DB427C9C0917" name="X7C18DB427C9C0917"></a></p> <div class="ChapSects"><a 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href="chap3.html#X8162E2BB7CF144F5">3 </div></div> </div> <h3>3 <span class="Heading"> Bipartitions and blocks </span></h3> <p>In this chapter we describe the functions in <strong class="pkg">Semigroups</strong> for creat <p>A <em>partition</em> of a set <span class="SimpleMath">X</span> is a set of pairwise disjoint non-em <p>Let <span class="SimpleMath">n∈</span><span class="SimpleMath">N</span> , let <span class="Simpl <p>The <em>partition monoid</em> of degree <span class="SimpleMath">n</span> is the set of all partiti <p>Let <span class="SimpleMath">x</span> be any bipartition of degree <span class="SimpleMath">n</ <p>Let <span class="SimpleMath">x</span> and <span class="SimpleMath">y</span> be bipartitions of d <p>Equivalently, the product <span class="SimpleMath">x</span> <span class="SimpleMath">y</span> <ul> <li><p><span class="SimpleMath">i</span> <code class="code">=</code><span class="SimpleMath">j</spa </li> <li><p><span class="SimpleMath">i</span> , <span class="SimpleMath">j</span> <span class="SimpleMat </li> <li><p><span class="SimpleMath">i</span> , <span class="SimpleMath">j</span> <span class="SimpleMat </li> </ul> <p>or there exists <span class="SimpleMath">r∈</span><span class="SimpleMath">N</span> and <span cla <ul> <li><p><span class="SimpleMath">r=2s-1</span> for some <span class="SimpleMath">s≥ 1</span> , <span cla <p class="pcenter">(i,-k(1))\in x,\ (k(1),k(2))\in y,\ (-k(2),-k(3))\in x,\ \ldots,\qquad</p> <p class="pcenter">\qquad\ldots,\ (-k(2s-2),-k(2s-1))\in x,\ (k(2s-1),j)\in y;</p> </li> <li><p><span class="SimpleMath">r=2s</span> for some <span class="SimpleMath">s≥ 1</span> , and either < <p class="pcenter">(i,-k(1))\in x,\ (k(1),k(2))\in y,\ (-k(2),-k(3))\in x,\ \ldots, (k(2s-1), k(2s))\in y,\ (-k(2s), j)\in x,</p> <p>or <span class="SimpleMath">i</span> ,<span class="SimpleMath">j</span> <span class="SimpleMat <p class="pcenter">(i,k(1))\in y,\ (-k(1),-k(2))\in x,\ (k(2),k(3))\in y,\ \ldots, (-k(2s-1), -k(2s))\in x,\ (k(2s), j)\in y.</p> </li> </ul> <p>This multiplication can be shown to be associative, and so the collection of all bipartitions <p>Let <span class="SimpleMath">x</span> be a bipartition of degree <span class="SimpleMath">n</sp <p class="pcenter"> (x^*)^*=x,\quad xx^*x=x,\quad x^*xx^*=x^*,\quad (xy)^*=y^*x^*. </p> <p>In this way, the partition monoid is a <em>regular *-semigroup</em>.</p> <p>A bipartition <span class="SimpleMath">x</span> of degree <span class="SimpleMath">n</span> is c <p>From a graphical perspective, as on Page 873 in <a href="chapBib.html#biBHalverson2005Part <p><a id="X7850845886902FBF" name="X7850845886902FBF"></a></p> <h4>3.1 <span class="Heading">The family and categories of bipartitions</span></h4> <p><a id="X80F11BEF856E7902" name="X80F11BEF856E7902"></a></p> <h5>3.1-1 IsBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p> <p>Every bipartition in <strong class="pkg">GAP</strong> belongs to the category <code class="cod <p><a id="X82F5D10C85489832" name="X82F5D10C85489832"></a></p> <h5>3.1-2 IsBipartitionCollection</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p> <p>Every collection of bipartitions belongs to the category <code class="code">IsBipartitionC <p>Every collection of collections of bipartitions belongs to <code class="code">IsBipartitio <p><a id="X85D77073820C7E72" name="X85D77073820C7E72"></a></p> <h4>3.2 <span class="Heading">Creating bipartitions</span></h4> <p>There are several ways of creating bipartitions in <strong class="pkg">GAP</strong>, which are <p><a id="X7E052E6378A5B758" name="X7E052E6378A5B758"></a></p> <h5>3.2-1 Bipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <p>Returns: A bipartition.</p> <p><code class="code">Bipartition</code> returns the bipartition <code class="code">x</code> with <p><code class="code">Bipartition</code> returns an error if the argument does not define a bipart <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -1], [2, 3, -3] <bipartition: [ 1, -1 ], [ 2, 3, -3 ], [ -2 ]></pre></div> <p><a id="X846AA7568435D2CE" name="X846AA7568435D2CE"></a></p> <h5>3.2-2 BipartitionByIntRep</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bi <p>Returns: A bipartition.</p> <p>It is possible to create a bipartition using its internal representation. The argument <var cl <p>For example, the internal representation of the bipartition with blocks</p> <div class="example"><pre>[1, -1], [2, 3, -2], [-3]</pre></div> <p>has internal representation</p> <div class="example"><pre>[1, 2, 2, 1, 2, 3]</pre></div> <p>The internal representation indicates that the number <code class="code">1</code> is in class <c <p>In its first form <code class="code">BipartitionByIntRep</code> verifies that the argument <va <p>See also <code class="func">IntRepOfBipartition</code> (<a href="chap3.html#X7ECD393A854C <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">BipartitionByIntRep([1, 2, 2, 1, 3, <bipartition: [ 1, -1 ], [ 2, 3 ], [ -2 ], [ -3 ]></pre></div> <p><a id="X8379B0538101FBC8" name="X8379B0538101FBC8"></a></p> <h5>3.2-3 IdentityBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Id <p>Returns: The identity bipartition.</p> <p>Returns the identity bipartition with degree <var class="Arg">n</var>.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">IdentityBipartition(10);</span> <block bijection: [ 1, -1 ], [ 2, -2 ], [ 3, -3 ], [ 4, -4 ], [ 5, -5 ], [ 6, -6 ], [ 7, -7 ], [ 8, -8 ], [ 9, -9 ], [ 10, -10 ]></pre></div> <p><a id="X824EDD4582AAA8C7" name="X824EDD4582AAA8C7"></a></p> <h5>3.2-4 LeftOne</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Returns: A bipartition.</p> <p>The <code class="code">LeftProjection</code> of a bipartition <var class="Arg">x</var> is the bip <p>The left projection <code class="code">e</code> of <var class="Arg">x</var> is also a bipartition w <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([</span> <span class="GAPprompt">></span> <span class="GAPinput"> [1, 4, -1, -2, -6], [2, 3, 5, -4], [6, -3], [ <span class="GAPprompt">gap></span> <span class="GAPinput">LeftOne(x);</span> <block bijection: [ 1, 4, -1, -4 ], [ 2, 3, 5, -2, -3, -5 ], [ 6, -6 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">LeftBlocks(x);</span> <blocks: [ 1*, 4* ], [ 2*, 3*, 5* ], [ 6* ]> <span class="GAPprompt">gap></span> <span class="GAPinput">RightBlocks(LeftOne(x));</span> <blocks: [ 1*, 4* ], [ 2*, 3*, 5* ], [ 6* ]> <span class="GAPprompt">gap></span> <span class="GAPinput">LeftBlocks(LeftOne(x));</span> <blocks: [ 1*, 4* ], [ 2*, 3*, 5* ], [ 6* ]> <span class="GAPprompt">gap></span> <span class="GAPinput">LeftOne(x) * x = x;</span> true</pre></div> <p><a id="X790B71108070FAC2" name="X790B71108070FAC2"></a></p> <h5>3.2-5 RightOne</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ri <p>Returns: A bipartition.</p> <p>The <code class="code">RightProjection</code> of a bipartition <var class="Arg">x</var> is the bi <p>The right projection <code class="code">e</code> of <var class="Arg">x</var> is also a bipartition <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -1, -4], [2, -2 <span class="GAPprompt">gap></span> <span class="GAPinput">RightOne(x);</span> <block bijection: [ 1, 4, -1, -4 ], [ 2, 3, -2, -3 ], [ 5, -5 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">RightBlocks(RightOne(x));</span> <blocks: [ 1*, 4* ], [ 2*, 3* ], [ 5* ]> <span class="GAPprompt">gap></span> <span class="GAPinput">LeftBlocks(RightOne(x));</span> <blocks: [ 1*, 4* ], [ 2*, 3* ], [ 5* ]> <span class="GAPprompt">gap></span> <span class="GAPinput">RightBlocks(x);</span> <blocks: [ 1*, 4* ], [ 2*, 3* ], [ 5* ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x * RightOne(x) = x;</span> true</pre></div> <p><a id="X7CE00E0C79F62745" name="X7CE00E0C79F62745"></a></p> <h5>3.2-6 StarOp</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A bipartition.</p> <p><code class="code">StarOp</code> returns the unique bipartition <code class="code">g</code> wit <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -4], [2, 3, 4], <bipartition: [ 1, -4 ], [ 2, 3, 4 ], [ 5 ], [ -1 ], [ -2, -3 ], [ -5 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">y := Star(x);</span> <bipartition: [ 1 ], [ 2, 3 ], [ 4, -1 ], [ 5 ], [ -2, -3, -4 ], [ -5 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x * y * x = x;</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">LeftBlocks(x) = RightBlocks(y);</s true <span class="GAPprompt">gap></span> <span class="GAPinput">RightBlocks(x) = LeftBlocks(y);</s true</pre></div> <p><a id="X8077265981409CCB" name="X8077265981409CCB"></a></p> <h5>3.2-7 RandomBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A bipartition.</p> <p>If <var class="Arg">n</var> is a positive integer, then <code class="code">RandomBipartition</c <p>If the optional first argument <var class="Arg">rs</var> is a random source, then this is used to g <p>Note that neither of these functions has a uniform distribution.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := RandomBipartition(6);</span> <bipartition: [ 1, 2, 3, 4 ], [ 5 ], [ 6, -2, -3, -4 ], [ -1, -5 ], [ -6 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x := RandomBlockBijection(4);</spa <block bijection: [ 1, 4, -2 ], [ 2, -4 ], [ 3, -1, -3 ]></pre></div> <p><a id="X7C2C44D281A0D2C9" name="X7C2C44D281A0D2C9"></a></p> <h4>3.3 <span class="Heading">Changing the representation of a bipartition</span></h4> <p>It is possible that a bipartition can be represented as another type of object, or that another <p>The operations <code class="func">AsPermutation</code> (<a href="chap3.html#X7C684CD38405 AsPartialPerm (3.3-4), AsTransformation (3.3-3) can be used to conv<p><a id="X855126D98583C181" name="X855126D98583C181"></a></p> <h5>3.3-1 AsBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A bipartition.</p> <p><code class="code">AsBipartition</code> returns the bipartition, permutation, transformat <p>There are several possible arguments for <code class="code">AsBipartition</code>:</p> <dl> <dt><strong class="Mark">permutations</strong></dt> <dd><p>If <var class="Arg">x</var> is a permutation and <var class="Arg">n</var> is a positive integer, t <p>If no positive integer <var class="Arg">n</var> is specified, then the largest moved point of <var </dd> <dt><strong class="Mark">transformations</strong></dt> <dd><p>If <var class="Arg">x</var> is a transformation and <var class="Arg">n</var> is a positive intege <p>If the positive integer <var class="Arg">n</var> is not specified, then the degree of <var class=" </dd> <dt><strong class="Mark">partial permutations</strong></dt> <dd><p>If <var class="Arg">x</var> is a partial permutation and <var class="Arg">n</var> is a positive in <p>If the optional argument <var class="Arg">n</var> is not present, then the default value of the ma </dd> <dt><strong class="Mark">bipartitions</strong></dt> <dd><p>If <var class="Arg">x</var> is a bipartition and <var class="Arg">n</var> is a non-negative integ <p>If <var class="Arg">n</var> equals the degree of <var class="Arg">x</var>, then <var class="Arg">x</v </dd> <dt><strong class="Mark">pbrs</strong></dt> <dd><p>If <var class="Arg">x</var> is a pbr satisfying <code class="func">IsBipartitionPBR</code> (<a h A returns the bipartition corresponding to x with degree n.</dd> </dl> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Transformation([3, 5, 3, 4, 1, 2] <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 5);</span> <bipartition: [ 1, 3, -3 ], [ 2, -5 ], [ 4, -4 ], [ 5, -1 ], [ -2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x);</span> <bipartition: [ 1, 3, -3 ], [ 2, -5 ], [ 4, -4 ], [ 5, -1 ], [ 6, -2 ], [ -6 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 10);</span> <bipartition: [ 1, 3, -3 ], [ 2, -5 ], [ 4, -4 ], [ 5, -1 ], [ 6, -2 ], [ 7, -7 ], [ 8, -8 ], [ 9, -9 ], [ 10, -10 ], [ -6 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition((1, 3)(2, 4));</span> <block bijection: [ 1, -3 ], [ 2, -4 ], [ 3, -1 ], [ 4, -2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition((1, 3)(2, 4), 10);</s <block bijection: [ 1, -3 ], [ 2, -4 ], [ 3, -1 ], [ 4, -2 ], [ 5, -5 ], [ 6, -6 ], [ 7, -7 ], [ 8, -8 ], [ 9, -9 ], [ 10, -10 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x := PartialPerm([1, 2, 3, 4, 5, 6], [6 <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 11);</span> <bipartition: [ 1, -6 ], [ 2, -7 ], [ 3, -1 ], [ 4, -4 ], [ 5, -3 ], [ 6, -2 ], [ 7 ], [ 8 ], [ 9 ], [ 10 ], [ 11 ], [ -5 ], [ -8 ], [ -9 ], [ -10 ], [ -11 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x);</span> <bipartition: [ 1, -6 ], [ 2, -7 ], [ 3, -1 ], [ 4, -4 ], [ 5, -3 ], [ 6, -2 ], [ 7 ], [ -5 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(Transformation([1 <block bijection: [ 1, -1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, 2, -2], [3], [4 <span class="GAPprompt">></span> <span class="GAPinput"> [-3, -4, -5, -6]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 0);</span> <empty bipartition> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 2);</span> <bipartition: [ 1, 2, -2 ], [ -1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 8);</span> <bipartition: [ 1, 2, -2 ], [ 3 ], [ 4, 5, 6, -1 ], [ 7 ], [ 8 ], [ -3, -4, -5, -6 ], [ -7 ], [ -8 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x := PBR(</span> <span class="GAPprompt">></span> <span class="GAPinput">[[-1, 1, 2, 3, 4], [-1, 1, 2, 3, 4],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [-1, 1, 2, 3, 4], [-1, 1, 2, 3, 4]],</span> <span class="GAPprompt">></span> <span class="GAPinput">[[-1, 1, 2, 3, 4], [-2], [-3], [-4]]);;</s <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x);</span> <bipartition: [ 1, 2, 3, 4, -1 ], [ -2 ], [ -3 ], [ -4 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 2);</span> <bipartition: [ 1, 2, -1 ], [ -2 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 4);</span> <bipartition: [ 1, 2, 3, 4, -1 ], [ -2 ], [ -3 ], [ -4 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 5);</span> <bipartition: [ 1, 2, 3, 4, -1 ], [ 5 ], [ -2 ], [ -3 ], [ -4 ], [ -5 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 0);</span> <empty bipartition></pre></div> <p><a id="X85A5AD2B7F3B776F" name="X85A5AD2B7F3B776F"></a></p> <h5>3.3-2 AsBlockBijection</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A block bijection.</p> <p>When the argument <var class="Arg">x</var> is a partial perm and <var class="Arg">n</var> is a positi <p>If the optional second argument <var class="Arg">n</var> is not present, then the maximum of the d <p>This is the value at <var class="Arg">x</var> of the embedding of the symmetric inverse monoid int <p>When the argument <var class="Arg">x</var> is a partial perm bipartition (see <code class="func"> AsBlockBijection(AsPartialPerm(x)[, n]).<div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := PartialPerm([1, 2, 3, 6, 7, 10], [ [2,5][3,6,1,9][10,8](7) <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(x, 11);</span> <bipartition: [ 1, -9 ], [ 2, -5 ], [ 3, -6 ], [ 4 ], [ 5 ], [ 6, -1 ], [ 7, -7 ], [ 8 ], [ 9 ], [ 10, -8 ], [ 11 ], [ -2 ], [ -3 ], [ -4 ], [ -10 ], [ -11 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">AsBlockBijection(x, 10);</span> Error, the 2nd argument (a pos. int.) is less than or equal to the max\ imum of the degree and codegree of the 1st argument (a partial perm) <span class="GAPprompt">gap></span> <span class="GAPinput">AsBlockBijection(x, 11);</span> <block bijection: [ 1, -9 ], [ 2, -5 ], [ 3, -6 ], [ 4, 5, 8, 9, 11, -2, -3, -4, -10, -11 ], [ 6, -1 ], [ 7, -7 ], [ 10, -8 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -3], [2], [3, - <span class="GAPprompt">gap></span> <span class="GAPinput">IsPartialPermBipartition(x);</sp true <span class="GAPprompt">gap></span> <span class="GAPinput">AsBlockBijection(x);</span> <block bijection: [ 1, -3 ], [ 2, 4, -1, -4 ], [ 3, -2 ]></pre></div> <p><a id="X7CE91D0C83865214" name="X7CE91D0C83865214"></a></p> <h5>3.3-3 AsTransformation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A transformation.</p> <p>When the argument <var class="Arg">x</var> is a bipartition, that mathematically defines a tran <p>See <code class="func">IsTransBipartition</code> (<a href="chap3.html#X79C556827A578509" <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -3], [2, -2], [ <span class="GAPprompt">></span> <span class="GAPinput"> [4, -12], [6, 7, -6], [8, -5], [9, -11],</s <span class="GAPprompt">></span> <span class="GAPinput"> [11, 12, -10], [-1], [-4], [-8], [-9]]); <span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation(x);</span> Transformation( [ 3, 2, 7, 12, 7, 6, 6, 5, 11, 7, 10, 10 ] ) <span class="GAPprompt">gap></span> <span class="GAPinput">IsTransBipartition(x);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, 5], [2, 4, 8, 10 <span class="GAPprompt">></span> <span class="GAPinput"> [3, 6, 7, -1, -2], [9, -4, -6, -9],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [-3, -5], [-7, -8], [-10]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">AsTransformation(x);</span> Error, the argument (a bipartition) does not define a transformation</pre></div> <p><a id="X7C5212EF7A200E63" name="X7C5212EF7A200E63"></a></p> <h5>3.3-4 AsPartialPerm</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A partial perm.</p> <p>When the argument <var class="Arg">x</var> is a bipartition that mathematically defines a parti <p>A bipartition <var class="Arg">x</var> defines a partial perm if and only if its numbers of left an <p>See <code class="func">IsPartialPermBipartition</code> (<a href="chap3.html#X87C771D37B1 <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -4], [2, -2], [ <span class="GAPprompt">></span> <span class="GAPinput"> [5, -9], [6], [7], [8, -6], [9, -3], [10, - <span class="GAPprompt">></span> <span class="GAPinput"> [-1], [-7]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsPartialPermBipartition(x);</sp true <span class="GAPprompt">gap></span> <span class="GAPinput">AsPartialPerm(x);</span> [1,4,5,9,3,10,8,6](2) <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -2, -4], [2, 3, <span class="GAPprompt">gap></span> <span class="GAPinput">IsPartialPermBipartition(x);</sp false <span class="GAPprompt">gap></span> <span class="GAPinput">AsPartialPerm(x);</span> Error, the argument (a bipartition) does not define a partial perm</pre></div> <p><a id="X7C684CD38405DBEF" name="X7C684CD38405DBEF"></a></p> <h5>3.3-5 AsPermutation</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A permutation.</p> <p>When the argument <var class="Arg">x</var> is a bipartition that mathematically defines a permu <p>A bipartition <var class="Arg">x</var> defines a permutation if and only if its numbers of left, r <p>See <code class="func">IsPermBipartition</code> (<a href="chap3.html#X8031B53E7D0ECCFA"> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, -6], [2, -4], [ <span class="GAPprompt">></span> <span class="GAPinput"> [5, -3], [6, -1]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsPermBipartition(x);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">AsPermutation(x);</span> (1,6)(2,4,5,3) <span class="GAPprompt">gap></span> <span class="GAPinput">AsBipartition(last) = x;</span> true</pre></div> <p><a id="X83F2C3C97E8FFA49" name="X83F2C3C97E8FFA49"></a></p> <h4>3.4 <span class="Heading">Operators for bipartitions</span></h4> <dl> <dt><strong class="Mark"><code class="code"><var class="Arg">f</var> * <var class="Arg">g</var></code>< <dd><p>returns the composition of <var class="Arg">f</var> and <var class="Arg">g</var> when <var class= </dd> <dt><strong class="Mark"><code class="code"><var class="Arg">f</var> < <var class="Arg">g</var></code></ <dd><p>returns <code class="keyw">true</code> if the internal representation of <var class="Arg">f</ </dd> <dt><strong class="Mark"><code class="code"><var class="Arg">f</var> = <var class="Arg">g</var></code>< <dd><p>returns <code class="keyw">true</code> if the bipartition <var class="Arg">f</var> equals the b </dd> </dl> <p><a id="X7A39D36086647536" name="X7A39D36086647536"></a></p> <h5>3.4-1 PartialPermLeqBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p> <p>If <var class="Arg">x</var> and <var class="Arg">y</var> are partial perm bipartitions, i.e. they s AsPartialPerm(x) < AsPartialPerm(y).<p><a id="X8608D78F83D55108" name="X8608D78F83D55108"></a></p> <h5>3.4-2 NaturalLeqPartialPermBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p> <p>The <em>natural partial order</em> <span class="SimpleMath">≤</span> on an inverse semigroup <code c <p><code class="code">NaturalLeqPartialPermBipartition</code> returns <code class="keyw">tru <p><a id="X79E8FA077E24C1F4" name="X79E8FA077E24C1F4"></a></p> <h5>3.4-3 NaturalLeqBlockBijection</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code>.</p> <p>The <em>natural partial order</em> <span class="SimpleMath">≤</span> on an inverse semigroup <code c <p><code class="code">NaturalLeqBlockBijection</code> returns <code class="keyw">true</code> if < <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, 2, -3], [3, -1, <span class="GAPprompt">></span> <span class="GAPinput"> [5, -5], [6, -6], [7, -7],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [8, -8], [9, -9], [10, -10]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">y := Bipartition([[1, -2], [2, -1], [ <span class="GAPprompt">></span> <span class="GAPinput"> [4, -4], [5, -5], [6, -6], [7, -7],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [8, -8], [9, -9], [10, -10]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">z := Bipartition([Union([1 .. 10], [ <span class="GAPprompt">gap></span> <span class="GAPinput">NaturalLeqBlockBijection(x, y);</ false <span class="GAPprompt">gap></span> <span class="GAPinput">NaturalLeqBlockBijection(y, x);</ false <span class="GAPprompt">gap></span> <span class="GAPinput">NaturalLeqBlockBijection(z, x);</ true <span class="GAPprompt">gap></span> <span class="GAPinput">NaturalLeqBlockBijection(z, y);</ true</pre></div> <p><a id="X7D9F5A248028FF52" name="X7D9F5A248028FF52"></a></p> <h5>3.4-4 PermLeftQuoBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A permutation.</p> <p>If <var class="Arg">x</var> and <var class="Arg">y</var> are bipartitions with equal left and right <p><code class="code">PermLeftQuoBipartition</code> verifies that <var class="Arg">x</var> and <va <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, 4, 6, 7, 8, 10], <span class="GAPprompt">></span> <span class="GAPinput"> [3, -3, -6, -7, -9], [9, -4, -5], [-10]]); <span class="GAPprompt">gap></span> <span class="GAPinput">y := Bipartition([[1, 4, 6, 7, 8, 10], <span class="GAPprompt">></span> <span class="GAPinput"> [3, -4, -5], [9, -1, -2, -8], [-10]]);;</s <span class="GAPprompt">gap></span> <span class="GAPinput">PermLeftQuoBipartition(x, y);</sp (1,2,3) <span class="GAPprompt">gap></span> <span class="GAPinput">Star(x) * y;</span> <bipartition: [ 1, 2, 8, -3, -6, -7, -9 ], [ 3, 6, 7, 9, -4, -5 ], [ 4, 5, -1, -2, -8 ], [ 10 ], [ -10 ]></pre></div> <p><a id="X87F3A304814797CE" name="X87F3A304814797CE"></a></p> <h4>3.5 <span class="Heading">Attributes for bipartitons</span></h4> <p>In this section we describe various attributes that a bipartition can possess.</p> <p><a id="X780F5E00784FE58C" name="X780F5E00784FE58C"></a></p> <h5>3.5-1 DegreeOfBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: A positive integer.</p> <p>The degree of a bipartition is, roughly speaking, the number of points where it is defined. Mor <p>The degree of a collection <var class="Arg">coll</var> of bipartitions of equal degree is just th <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">x := Bipartition([[1, 7, -3, -8], [2, <span class="GAPprompt">></span> <span class="GAPinput"> [3], [4, -7, -9], [5, 9, -2],</span> <span class="GAPprompt">></span> <span class="GAPinput"> [8, -1, -4, -6], [-5]]);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfBipartition(x);</span> 9 <span class="GAPprompt">gap></span> <span class="GAPinput">S := BrauerMonoid(5);</span> <regular bipartition *-monoid of degree 5 with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">IsBipartitionCollection(S);</spa true <span class="GAPprompt">gap></span> <span class="GAPinput">DegreeOfBipartitionCollection(S 5</pre></div> <p><a id="X82074756826AD2C2" name="X82074756826AD2C2"></a></p> <h5>3.5-2 RankOfBipartition</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">& <p>Returns: The rank of a bipartition.</p> --> -------------------- --> maximum size reached --> --------------------
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