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<p><a id="X7D2C85EC87DD46E5" name="X7D2C85EC87DD46E5"></a></p>
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<h1><strong class="pkg">Gpd</strong></h1>
<h2>Groupoids, graphs of groups, and graphs of groupoids</h2>
<p>Version 1.05</p>
<p>November 2008</p>
</div>
<p><b> Emma Moore
</b>
<br />Email: <span class="URL"><a href="mailto:emmajmoore@yahoo.co.uk">emmajmoore@yahoo.co.uk</a></span>
</p><p><b> Chris Wensley
</b>
<br />Email: <span class="URL"><a href="mailto:c.d.wensley@bangor.ac.uk">c.d.wensley@bangor.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="http://www.bangor.ac.uk/~mas023/">http://www.bangor.ac.uk/~mas023/</a></span>
<br />Address: <br />School of Computer Science, Bangor University,<br /> Dean Street, Bangor, Gwynedd, LL57 1UT, U.K.
</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>The <strong class="pkg">Gpd</strong> package for <strong class="pkg">GAP</strong>4 provides functions for the computation with groupoids (categories with every arrow invertible) and their morphisms; for graphs of groups, and graphs of groupoids.</p>
<p>It provides normal forms for Free Products with Amalgamation and for HNN-extensions when the initial groups have rewrite systems and the subgroups have finite index.</p>
<p>The <strong class="pkg">Gpd</strong> package was originally implemented in 2000 (as <strong class="pkg">GraphGpd</strong>) when the first author was studying for a Ph.D. in Bangor.</p>
<p>The current version is 1.05, released on 21st November 2008, and now includes the more basic structure of <em>magma with objects</em>. It had to be released hurriedly, due to the change of website, so some of the function are no longer available. A new version will be released as soon as possible.</p>
<p>Bug reports, suggestions and comments are, of course, welcome. Please contact the second author at <span class="URL"><a href="mailto:c.d.wensley@bangor.ac.uk">c.d.wensley@bangor.ac.uk</a></span>.</p>
<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2000-2008 Emma Moore and Chris Wensley</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This <strong class="pkg">gpd</strong> package is released under the GNU General Public License (GPL). This file is part of <strong class="pkg">gpd</strong>, though as documentation it is released under the GNU Free Documentation License (see <span class="URL"><a href="http://www.gnu.org/licenses/licenses.html#FDL">http://www.gnu.org/licenses/licenses.html#FDL</a></span>).</p>
<p><strong class="pkg">gpd</strong> is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version.</p>
<p><strong class="pkg">gpd</strong> is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.</p>
<p>You should have received a copy of the GNU General Public License along with <strong class="pkg">gpd</strong>; if not, write to the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.</p>
<p>For more details, see <span class="URL"><a href="http://www.fsf.org/licenses/gpl.html">http://www.fsf.org/licenses/gpl.html</a></span>.</p>
<p>This documentation was prepared with the <strong class="pkg">GAPDoc</strong> package of Frank L\"ubeck and Max Neunh\"offer.</p>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>
<div class="contents">
<h3>Contents</h3>
<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
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<div class="ContChap"><a href="chap2.html#X8361CEA4856430C6">2 <span class="Heading">Many-object structures</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X79CE5DFF7ADC7744">2.1 <span class="Heading">Magmas with objects</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C51D7847BD23284">2.1-1 MagmaWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X79D2548F7DD8C6E7">2.1-2 MultiplicativeElementWithObjects</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7C0911667FF90DB2">2.1-3 IsSinglePiece</a></span>
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</div>
<div class="ContChap"><a href="chap3.html#X85297B407ACAED81">3 <span class="Heading">Homomorphisms of many-object structures</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X82F856A086B93832">3.1 <span class="Heading">Homomorphisms of magmas with objects</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86E00FEA7FF38FEA">3.1-1 MagmaWithObjectsHomomorphism</a></span>
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</div>
<div class="ContChap"><a href="chap4.html#X82F6A1AB798648F4">4 <span class="Heading">Groupoids</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7A7355C58610A48C">4.1 <span class="Heading">Groupoids: their elements and attributes</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8406913B7ED86CFE">4.1-1 SinglePieceGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8382678F86F18DA8">4.1-2 Pieces</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8511D3EE845CC930">4.1-3 IsPermGroupoid</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D028B3B8385ED07">4.1-4 GroupoidElement</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C0911667FF90DB2">4.1-5 IsSinglePiece</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7BDBA72C852C4625">4.2 <span class="Heading">Subgroupoids</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7AEA9BF780CD6957">4.2-1 SubgroupoidByPieces</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X793A3BBE81410B11">4.3 <span class="Heading">Stars, Costars and Homsets</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B561BAE7D471C60">4.3-1 ObjectStar</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E30871D78B9F81C">4.3-2 IdentityElement</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X831AA9E8780235F2">4.4 <span class="Heading">Left, right and double cosets</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8412ABD57986B9FC">4.4-1 RightCoset</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X8653FC9786E3209A">4.5 <span class="Heading">Conjugation</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8123456781234567">4.5-1 \^</a></span>
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<div class="ContChap"><a href="chap5.html#X879ED2A7878B5A7A">5 <span class="Heading">Homomorphisms of Groupoids</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7EAAB2A97E097FB6">5.1 <span class="Heading">Homomorphisms to a connected groupoid</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X8013645279DEEB6F">5.1-1 GroupoidHomomorphism</a></span>
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<div class="ContChap"><a href="chap6.html#X78063DC8847554B4">6 <span class="Heading">Graphs of Groups and Groupoids</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7D554C5D7FDC3D02">6.1 <span class="Heading">Digraphs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X85BD6D2584D8A22F">6.1-1 FpWeightedDigraph</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7BAFCA3680E478AE">6.2 <span class="Heading">Graphs of Groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8130246E854BC5D9">6.2-1 GraphOfGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X847464677F641527">6.2-2 IsGraphOfFpGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7B036C2B84E48BB1">6.2-3 RightTransversalsOfGraphOfGroups</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7BD9DCF87FB3E0AF">6.3 <span class="Heading">Words in a Graph of Groups and their normal forms</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X87937AE47C5B1018">6.3-1 GraphOfGroupsWord</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X78FA7C3F831AA6E4">6.3-2 ReducedGraphOfGroupsWord</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X7D99A7B37B36BAA8">6.4 <span class="Heading">Free products with amalgamation and HNN extensions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X795AB71F7E370119">6.4-1 FreeProductWithAmalgamation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7CB0F120804A8DED">6.4-2 HnnExtension</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap6.html#X78108FB4814AE887">6.5 <span class="Heading">GraphsOfGroupoids and their Words</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8739267678808E85">6.5-1 GraphOfGroupoids</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7EC1A9067FC91255">6.5-2 GraphOfGroupoidsWord</a></span>
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<div class="ContChap"><a href="chap7.html#X810C43BC7F63C4B4">7 <span class="Heading">Development History</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X8192EA4C7B7CC5CD">7.1 <span class="Heading">Versions of the Package</span></a>
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<div class="ContSect"><span class="nocss"> </span><a href="chap7.html#X82148FB77CC3CCC0">7.2 <span class="Heading">What needs to be done next?</span></a>
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