<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>
<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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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 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>Pregroup of a free group of rank <span class="SimpleMath">\(k\)</span>. The only defined products are <span class="SimpleMath">\(1\cdot x = x \cdot 1 = x\)</span> and <span class="SimpleMath">\(xx^{-1} = x^{-1}x = 1\)</span>, for all generators <span class="SimpleMath">\(x\)</span>.</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ []</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ 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>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>
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