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<h1><strong class="pkg">IdRel</strong></h1>
<h2>A package for Identities among Relators</h2>
<p>Version 2.05</p>
<p>November 2008</p>
</div>
<p><b> Anne Heyworth
</b>
<br />Email: <span class="URL"><a href="mailto:anne.heyworth@googlemail.com">anne.heyworth@googlemail.com</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">IdRel</strong> package was originally implemented in 1999, using the <strong class="pkg">GAP</strong> 3 language, when the first author was studying for a Ph.D. in Bangor.</p>
<p>This package is designed to compute a minimal set of generators for the module of the identities among relators of a group presentation. It does this using</p>
<ul>
<li><p>rewriting and logged rewriting: a self-contained implementation of the Knuth-Bendix process using the monoid presentation associated to the group presentation;</p>
</li>
<li><p>monoid polynomials: an implementation of the monoid ring;</p>
</li>
<li><p>module polynomials: an implementation of the right module over this monoid generated by the relators.</p>
</li>
<li><p>Y-sequences: used as a <em>rewriting</em> way of representing elements of a free crossed module (products of conjugates of group relators and inverse relators).</p>
</li>
</ul>
<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>© 2005-2008 Anne Heyworth and Chris Wensley</p>
<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>This <strong class="pkg">idrel</strong> package is released under the GNU General Public License (GPL). This file is part of <strong class="pkg">idrel</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">idrel</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">idrel</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">idrel</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>
</div>
<div class="ContChap"><a href="chap2.html#X7CA8FCFD81AA1890">2 <span class="Heading">Rewriting Systems</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7C9AE9BC78CCBFAA">2.1 <span class="Heading">Identity Y-sequences</span></a>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7875619E84157FC1">2.2 <span class="Heading">Monoid Presentations of FpGroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X868422B878B0C380">2.2-1 FreeRelatorGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7CBE13927DFF4446">2.2-2 MonoidPresentationFpGroup</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X7A1F10597D8FC9A9">2.3 <span class="Heading">Rewriting systems for FpGroups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X858ECE3E807C7363">2.3-1 RewritingSystemFpGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X83BD6C0A80D88C2C">2.3-2 OnePassReduceWord</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7F0CD1EB7C220D40">2.3-3 OnePassKB</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap2.html#X83CBF2BE8478A728">2.4 <span class="Heading">Enumerating elements</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2.html#X7EDA50068207339D">2.4-1 ElementsOfMonoidPresentation</a></span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X7B8D727485966AF8">3 <span class="Heading">Logged Rewriting Systems</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X797732E87F1FE197">3.1 <span class="Heading">Logged Knuth-Bendix Completion</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X80075D5180A8F1A5">3.1-1 LoggedOnePassKB</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X87D1E3A578AAAFCB">3.1-2 LoggedKnuthBendix</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chap3.html#X831A93087918AA5D">3.2 <span class="Heading">Logged reduction of a word</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7C5094AF784A8BA7">3.2-1 LoggedReduceWordKB</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X8652CEEF7802DA46">3.2-2 LoggedRewritingSystemFpGroup</a></span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X83B25026816C87CE">4 <span class="Heading">Monoid Polynomials</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7E8CE44085FE959D">4.1 <span class="Heading">Construction of monoid polynomials</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DE231F282DB8660">4.1-1 MonoidPolyFromCoeffsWords</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X84B1F4CC79983A94">4.2 <span class="Heading">Components of a polynomial</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X832D20E8813FBE5D">4.2-1 Terms</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85C78946877C1FF5">4.2-2 Monic</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82E04F1086DAAA43">4.2-3 AddTermMonoidPoly</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X832341AB7A04BA45">4.3 <span class="Heading">Monoid Polynomial Operations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X780769238600AFD1">4.3-1 Length</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7A80EA5C7AEA68B1">4.4 <span class="Heading">Reduction of a Monoid Polynomial</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7979DE308676398D">4.4-1 ReduceMonoidPoly</a></span>
</div>
</div>
<div class="ContChap"><a href="chap5.html#X7B5CEEDF82747121">5 <span class="Heading">Module Polynomials</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X86625AB980F24AA5">5.1 <span class="Heading">Construction of module polynomials</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BEE34B27861ACE5">5.1-1 ModulePoly</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X83ECC2D5781DE850">5.2 <span class="Heading">Components of a module polynomial</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X832D20E8813FBE5D">5.2-1 Terms</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X7E57DFF4791C4CAA">5.3 <span class="Heading">Module Polynomial Operations</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X811C7964873E4062">5.3-1 AddTermModulePoly</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap5.html#X78038BF07E998E21">5.4 <span class="Heading">Identities among relators</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X78A94CB77B98ACAA">5.4-1 IdentityYSequences</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap5.html#X7BEE0DBB78F9355E">5.4-2 RootIdentities</a></span>
</div>
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