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<div class="ChapSects"><a href="chap2.html#X84CF717F7CE7F8A4">2 <span class="Heading">Pregroups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7F44CAE5832B8511">2.1 <span class="Heading">Creating Pregroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7E37DF907C05522C">2.1-1 PregroupByTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8137D096798C1BC8">2.1-2 PregroupByRedRelators</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8632A365791D87B7">2.1-3 PregroupOfFreeProduct</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8344F038858426A6">2.1-4 PregroupOfFreeGroup</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B1844C180E739B9">2.2 <span class="Heading">Filters and Representations</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86AA05DA7B7BA64B">2.2-1 IsPregroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X78E417B3823387C2">2.2-2 IsPregroupTableRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84F97B4D79FBAD44">2.2-3 IsPregroupOfFreeGroupRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X82866B1A7D018BB7">2.2-4 IsPregroupOfFreeProductRep</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8053940287FA4077">2.3 <span class="Heading">Attributes, Properties, and Operations</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7946D406823A935B">2.3-1 []</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80EC827C83DB818B">2.3-2 IntermultPairs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C1AE0D983EF9693">2.3-3 One</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X8761B5A57CC4DEA8">2.3-4 MultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7BF3B344878B252F">2.3-5 SetPregroupElementNames</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X82B7E2227A94AC98">2.3-6 PregroupElementNames</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7E4700087FC3F933">2.4 <span class="Heading">Elements of Pregroups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7CACF96A831CEB1F">2.4-1 IsElementOfPregroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X80185F7579C23714">2.4-2 IsElementOfPregroupRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X838ECC967A2377BA">2.4-3 IsElementOfPregroupOfFreeGroupRep</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X784ACAE2800C3E7E">2.4-4 PregroupOf</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X84A42E6B7D542AD1">2.4-5 IsDefinedMultiplication</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X86FC5FDF87C3664A">2.4-6 IsIntermultPair</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X87004CD07832F9DB">2.4-7 PregroupInverse</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X870A0A6A7CAE4680">2.5 <span class="Heading">Small Pregroups</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7AAEA96C79A55FB3">2.5-1 NrSmallPregroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7FD30E6F812704C8">2.5-2 SmallPregroup</a></span>
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<h3>2 <span class="Heading">Pregroups</span></h3>
<p>Pregroups are the fundamental building block of pregroup presentations used in the hyperbolicity tester.</p>
<p><a id="X7F44CAE5832B8511" name="X7F44CAE5832B8511"></a></p>
<h4>2.1 <span class="Heading">Creating Pregroups</span></h4>
<p>This section describes functions to create pregroups from multiplication tables, free groups, and free products of finite groups.</p>
<p><a id="X7E37DF907C05522C" name="X7E37DF907C05522C"></a></p>
<h5>2.1-1 PregroupByTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupByTable</code>( <var class="Arg">enams</var>, <var class="Arg">table</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupByTableNC</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: A pregroup</p>
<p>If <var class="Arg">enams</var> is a list of element names, which can be arbitrary GAP objects, with the convention that <code class="code">enams[1]</code> is the name of the identity element, and <var class="Arg">table</var> is a square table of non-negative integers that is the multiplication table of a pregroup, then <code class="func">PregroupByTable</code> and <code class="func">PregroupByTableNC</code> return a pregroup in multiplication table representation.</p>
<p>By convention the elements of the pregroup are numbered <code class="code">[1..n]</code> with <code class="code">0</code> denoting an undefined product in the table.</p>
<p>The axioms for a pregroup are checked by <code class="func">PregroupByTable</code> and not checked by <code class="func">PregroupByTableNC</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pregroup := PregroupByTable( "1xyY",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ [1,2,3,4]</span>
<span class="GAPprompt">></span> <span class="GAPinput"> , [2,1,0,0]</span>
<span class="GAPprompt">></span> <span class="GAPinput"> , [3,4,0,1]</span>
<span class="GAPprompt">></span> <span class="GAPinput"> , [4,0,1,3] ] );</span>
<pregroup with 4 elements in table rep>
</pre></div>
<p><a id="X8137D096798C1BC8" name="X8137D096798C1BC8"></a></p>
<h5>2.1-2 PregroupByRedRelators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupByRedRelators</code>( <var class="Arg">F</var>, <var class="Arg">rrel</var>, <var class="Arg">inv</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A pregroup in table representation</p>
<p>Construct a pregroup from the list <var class="Arg">rrel</var> of red relators and the list <var class="Arg">inv</var> of involutions over the free group <var class="Arg">F</var>. The argument <var class="Arg">rred</var> has to be a list of elements of length 3 in the free group <var class="Arg">F</var>, and <var class="Arg">inv</var> has to be a list of generators of <var class="Arg">F</var>.</p>
<p><a id="X8632A365791D87B7" name="X8632A365791D87B7"></a></p>
<h5>2.1-3 PregroupOfFreeProduct</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupOfFreeProduct</code>( <var class="Arg">G</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Construct the pregroup of the free product of <var class="Arg">G</var> and <var class="Arg">H</var>. If <var class="Arg">G</var> and <var class="Arg">H</var> are finite groups, then <code class="func">PregroupOfFreeProduct</code> returns the pregroup consisting of the non-identity elements of <var class="Arg">G</var> and <var class="Arg">H</var> and an identity element. A product between two non-trivial elements is defined if and only if they are in the same group.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">pregroup := PregroupOfFreeProduct(SmallGroup(12,2), SmallGroup(24,4));</span>
<pregroup with 35 elements in table rep>
</pre></div>
<p><a id="X8344F038858426A6" name="X8344F038858426A6"></a></p>
<h5>2.1-4 PregroupOfFreeGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupOfFreeGroup</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Return the pregroup of the free group <var class="Arg">F</var></p>
<p><a id="X7B1844C180E739B9" name="X7B1844C180E739B9"></a></p>
<h4>2.2 <span class="Heading">Filters and Representations</span></h4>
<p>This section gives an overview over the filters, categories and representations defined by <strong class="pkg">walrus</strong></p>
<p><a id="X86AA05DA7B7BA64B" name="X86AA05DA7B7BA64B"></a></p>
<h5>2.2-1 IsPregroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPregroup</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p><a id="X78E417B3823387C2" name="X78E417B3823387C2"></a></p>
<h5>2.2-2 IsPregroupTableRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPregroupTableRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p>A pregroup represented by its multiplication table, which is a square table of integers between 0 and the size of the pregroup, where 0 represents an undefined multiplication.</p>
<p><a id="X84F97B4D79FBAD44" name="X84F97B4D79FBAD44"></a></p>
<h5>2.2-3 IsPregroupOfFreeGroupRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPregroupOfFreeGroupRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p>Pregroup of a free group of rank <span class="SimpleMath">k</span>. The only defined products are <span class="SimpleMath">1⋅ x = x ⋅ 1 = x</span> and <span class="SimpleMath">xx^-1 = x^-1x = 1</span>, for all generators <span class="SimpleMath">x</span>.</p>
<p><a id="X82866B1A7D018BB7" name="X82866B1A7D018BB7"></a></p>
<h5>2.2-4 IsPregroupOfFreeProductRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsPregroupOfFreeProductRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p>Pregroup of the free product of a list of groups where products between non-trivial elements <span class="SimpleMath">g</span>, <span class="SimpleMath">h</span> are defined if <span class="SimpleMath">g,h</span> are contained in the same group.</p>
<p><a id="X8053940287FA4077" name="X8053940287FA4077"></a></p>
<h4>2.3 <span class="Heading">Attributes, Properties, and Operations</span></h4>
<p>This section gives an overview over the attributes, properties, and operatins defined for pregroups.</p>
<p><a id="X7946D406823A935B" name="X7946D406823A935B"></a></p>
<h5>2.3-1 []</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; []</code>( <var class="Arg">pregroup</var>, <var class="Arg">i</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Get the <var class="Arg">i</var>th element of <var class="Arg">pregroup</var>. By convention the <span class="SimpleMath">1</span>st element is the identity element.</p>
<p><a id="X80EC827C83DB818B" name="X80EC827C83DB818B"></a></p>
<h5>2.3-2 IntermultPairs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IntermultPairs</code>( <var class="Arg">pregroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns the set of intermult pairs of the pregroup</p>
<p><a id="X7C1AE0D983EF9693" name="X7C1AE0D983EF9693"></a></p>
<h5>2.3-3 One</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; One</code>( <var class="Arg">pregroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The identity element of <var class="Arg">pregroup</var>.</p>
<p><a id="X8761B5A57CC4DEA8" name="X8761B5A57CC4DEA8"></a></p>
<h5>2.3-4 MultiplicationTable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; MultiplicationTable</code>( <var class="Arg">pregroup</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The multiplication table of <var class="Arg">pregroup</var></p>
<p><a id="X7BF3B344878B252F" name="X7BF3B344878B252F"></a></p>
<h5>2.3-5 SetPregroupElementNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SetPregroupElementNames</code>( <var class="Arg">pregroup</var>, <var class="Arg">names</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Can be used to set more user-friendly display names for the elements of <var class="Arg">pregroup</var>. The list <var class="Arg">names</var> has to be of length <code class="code">Size(<var class="Arg">pregroup</var>)</code>.</p>
<p><a id="X82B7E2227A94AC98" name="X82B7E2227A94AC98"></a></p>
<h5>2.3-6 PregroupElementNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupElementNames</code>( <var class="Arg">pregroup</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Return the list of names of elements of <var class="Arg">pregroup</var></p>
<p><a id="X7E4700087FC3F933" name="X7E4700087FC3F933"></a></p>
<h4>2.4 <span class="Heading">Elements of Pregroups</span></h4>
<p><a id="X7CACF96A831CEB1F" name="X7CACF96A831CEB1F"></a></p>
<h5>2.4-1 IsElementOfPregroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsElementOfPregroup</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p><a id="X80185F7579C23714" name="X80185F7579C23714"></a></p>
<h5>2.4-2 IsElementOfPregroupRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsElementOfPregroupRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p><a id="X838ECC967A2377BA" name="X838ECC967A2377BA"></a></p>
<h5>2.4-3 IsElementOfPregroupOfFreeGroupRep</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsElementOfPregroupOfFreeGroupRep</code>( <var class="Arg">arg</var> )</td><td class="tdright">( filter )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>
<p><a id="X784ACAE2800C3E7E" name="X784ACAE2800C3E7E"></a></p>
<h5>2.4-4 PregroupOf</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupOf</code>( <var class="Arg">p</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The pregroup that the element <var class="Arg">p</var> is contained in.</p>
<p><a id="X84A42E6B7D542AD1" name="X84A42E6B7D542AD1"></a></p>
<h5>2.4-5 IsDefinedMultiplication</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsDefinedMultiplication</code>( <var class="Arg">p</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Tests whether the multiplication of <var class="Arg">p</var> and <var class="Arg">q</var> is defined in the pregroup containing <var class="Arg">p</var> and <var class="Arg">q</var>.</p>
<p><a id="X86FC5FDF87C3664A" name="X86FC5FDF87C3664A"></a></p>
<h5>2.4-6 IsIntermultPair</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsIntermultPair</code>( <var class="Arg">p</var>, <var class="Arg">q</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Tests whether <span class="SimpleMath">(<var class="Arg">p</var>, <var class="Arg">q</var>)</span> is an intermult pair. defined.</p>
<p><a id="X87004CD07832F9DB" name="X87004CD07832F9DB"></a></p>
<h5>2.4-7 PregroupInverse</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; PregroupInverse</code>( <var class="Arg">p</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Return the inverse of <var class="Arg">p</var>.</p>
<p><a id="X870A0A6A7CAE4680" name="X870A0A6A7CAE4680"></a></p>
<h4>2.5 <span class="Heading">Small Pregroups</span></h4>
<p>This package contains a small database of pregroups of sizes <span class="SimpleMath">1</span> to <span class="SimpleMath">7</span>. The database was computed by Chris Jefferson using the Minion constraint solver.</p>
<p>These small pregroups currently used for testing. Accessing the small pregroups database works as follows.</p>
<p><a id="X7AAEA96C79A55FB3" name="X7AAEA96C79A55FB3"></a></p>
<h5>2.5-1 NrSmallPregroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NrSmallPregroups</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: an integer.</p>
<p>Returns the number of pregroups of size <var class="Arg">n</var> available in the database.</p>
<p><a id="X7FD30E6F812704C8" name="X7FD30E6F812704C8"></a></p>
<h5>2.5-2 SmallPregroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SmallPregroup</code>( <var class="Arg">n</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a pregroup.</p>
<p>Returns the <var class="Arg">i</var>th pregroup of size <var class="Arg">n</var> from the database of small pregroups.</p>
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2026-11-01
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