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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a>  <a href="chap1.html">1</a>  <a href="chap2.html">2</a>  <a href="chap3.html">3</a>  <a href="chap4.html">4</a>  <a href="chap5.html">5</a>  <a href="chapA.html">A</a>  <a href="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

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<h1>GBNP</h1>


<h2>computing Gröbner bases of noncommutative polynomials</h2>

<p>
    1.1.0</p>

<p>
    29 August 2024
  </p>

</div>
<p><b>
    A.M. Cohen


  </b>
<br />Email: <span class="URL"><a href="mailto:A.M.Cohen@tue.nl">A.M.Cohen@tue.nl</a></span>
</p><p><b>
    J.W. Knopper


  </b>
<br />Email: <span class="URL"><a href="mailto:J.W.Knopper@tue.nl">J.W.Knopper@tue.nl</a></span>
</p>

<p><b>Address:</b><br />
TU/e,<br /> POB 513, 5600 MB Eindhoven, the Netherlands</p>
<p><a id="X7AA6C5737B711C89" name="X7AA6C5737B711C89"></a></p>
<h3>Abstract</h3>
<p>We provide algorithms, written in the <strong class="pkg">GAP</strong> 4 programming language, for computing Gröbner bases of non-commutative polynomials, and some variations, such as a weighted and truncated version and a tracing facility. In addition, there are algorithms for analyzing the quotient of a non-commutative polynomial algebra by a 2-sided ideal generated by a set of polynomials whose Gröbner basis has been determined and for computing quotient modules of free modules over quotient algebras.</p>

<p>The notion of algorithm is interpreted loosely: in general one cannot expect a non-commutative Gröbner basis algorithm to terminate, as it would imply solvability of the word problem for finitely presented (semi)groups.</p>

<p>This documentation gives a short description of the mathematical content, explains the functions of the package, and provides more than twenty worked out examples.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group at the Department of Mathematics and Computer Science of Eindhoven University of Technology.</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>

<ul>
<li><p>The package is based on an earlier version by Rosane Ushirobira.</p>

</li>
<li><p>The bulk of the package is written by Arjeh M. Cohen and Dié A.H. Gijsbers.</p>

</li>
<li><p>The theory is mainly taken from literature by Teo Mora <a href="chapBib.html#biBTCS::Mora1994:131">[Mor94]</a> and Edward L. Green <a href="chapBib.html#biBGreen1997">[Gre99]</a>.</p>

</li>
<li><p>From Version 0.8.3 on the package has three additional files (<code class="file">fincheck.g</code>, <code class="file">tree.g</code> <code class="file">graphs.g</code>) with routines for finding the Hilbert function and testing finite dimensionality when given a Gröbner basis by Chris Krook <a href="chapBib.html#biBKrook2003">[Kro03]</a>, based on work by Victor Ufnarovski <a href="chapBib.html#biBMR91d:16053">[Ufn89]</a>.</p>

</li>
<li><p>From Version 0.9 on the package is enriched with support for fields implemented in GAP and additional prefix rules for quotient modules, as well as some speed improvements by Jan Willem Knopper. Knopper has also formatted the documentation in GAPDoc <a href="chapBib.html#biBGAPDoc">[LN06]</a>.</p>

</li>
<li><p>From Version 1.0 on the package is extended with NMO (for Noncommutative Monomial Orderings) by Randall Cone. This enables the GBNP user to choose a wider selection of monomial orderings than the standard one built into GBNP itself. Documentation on NMO can be found in the NMO manual <a href="chapBib.html#biBNMODoc">[Con10]</a>.</p>

</li>
</ul>
<p><a id="X8537FEB07AF2BEC8" name="X8537FEB07AF2BEC8"></a></p>

<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8360C04082558A12">1.1 <span class="Heading">Installation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X78629CD778BE8C5D">1.2 <span class="Heading">Using the package</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DDEF24284C861D8">1.3 <span class="Heading">Further documentation</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X7BBCB13F82ACC213">2 <span class="Heading">Description</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7FDF3E5E7F33D3A2">2.1 <span class="Heading">Non-commutative Polynomials (NPs)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7B27E2D1784538DE">2.2 <span class="Heading">Non-commutative Polynomials for Modules (NPMs)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X84BD98F5811EAC45">2.3 <span class="Heading">Core functions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7EEE260680A64013">2.4 <span class="Heading">About the implementation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X8739B6547BC89505">2.5 <span class="Heading">Tracing
variant</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X78CF5C44879D34B6">2.6 <span class="Heading">Truncation variant</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86F1F4EE7D4D06B7">2.7 <span class="Heading">Module variant</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X80DAE0A97CFC95DD">2.8 <span class="Heading">Gröbner basis records</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X85A91A467FF1DE45">2.9 <span class="Heading">Quotient algebras</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X86FA580F8055B274">3 <span class="Heading">Functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81ABB91B79E00229">3.1 <span class="Heading">Converting polynomials
            into different formats</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B0EBCBC7857F1AE">3.1-1 GP2NP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CF0ED937DDA5A7E">3.1-2 GP2NPList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X86C3912F781ABEDC">3.1-3 NP2GP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X844A23EA7D97150C">3.1-4 NP2GPList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X78F44B01851B1020">3.2 <span class="Heading">Printing polynomials in NP format</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7B63BEA87A8D6162">3.2-1 PrintNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F7510A878045D3A">3.2-2 GBNP.ConfigPrint</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X832103DC79A9E9D0">3.2-3 PrintNPList</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X83DE3F817EA74727">3.3 <span class="Heading">Calculating with polynomials in NP format</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DB3792385AAA805">3.3-1 NumAlgGensNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X865548F07C74AB0A">3.3-2 NumAlgGensNPList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X782647C57D148379">3.3-3 NumModGensNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8119282084CA8076">3.3-4 NumModGensNPList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X788E1ACA82A833A8">3.3-5 AddNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84FC611A822D808F">3.3-6 BimulNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X855F3D4C783000E3">3.3-7 CleanNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D05B60E83FDA567">3.3-8 GtNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8075AE7E7A8088FF">3.3-9 LtNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A42AE79811CC5D7">3.3-10 LMonNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80CD462F794A8095">3.3-11 LTermNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X878A8C027DA25196">3.3-12 MkMonicNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X818147CD841BD490">3.3-13 FactorOutGcdNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7ABA720E87EFF040">3.3-14 MulNP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81381B2D83D2B9A9">3.4 <span class="Heading">Gröbner functions, standard variant</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CD9F9C97B2563E2">3.4-1 Grobner</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FEDA29E78B0CEED">3.4-2 SGrobner</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80D4D22C7E643C7B">3.4-3 IsGrobnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7D17F9027F08CF0B">3.4-4 IsStrongGrobnerBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E0105ED7FF4210F">3.4-5 IsGrobnerPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8752DA1A7CAF77D3">3.4-6 MakeGrobnerPair</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7F387F7780425B9A">3.5 <span class="Heading">Finite-dimensional quotient
        algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7EAA04247B2C6330">3.5-1 BaseQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X81A50EEE7B56C723">3.5-2 DimQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DFA841A8425DD94">3.5-3 MatrixQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78E4BF2F7F0D5E74">3.5-4 MatricesQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X80C4D0E882B05FDF">3.5-5 MulQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8563683E7FA604F8">3.5-6 StrongNormalFormNP</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X79FE4A3983E2329F">3.6 <span class="Heading">Finiteness and Hilbert series</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83C57C3A7DCF0471">3.6-1 DetermineGrowthQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X792E39A98717D779">3.6-2 FinCheckQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CFD47367CF309EB">3.6-3 HilbertSeriesQA</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X863124677B933CEE">3.6-4 PreprocessAnalysisQA</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7BA5CAA07890F7AA">3.7 <span class="Heading">Functions of the
            trace variant</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X813454F6799B1D57">3.7-1 EvalTrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83D1560C7F2A04BA">3.7-2 PrintTraceList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8039BEE77C070FB1">3.7-3 PrintTracePol</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7DD0B56D7BD6CD98">3.7-4 PrintNPListTrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X78AE6EED83B97595">3.7-5 SGrobnerTrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8219059A86A54130">3.7-6 StrongNormalFormTraceDiff</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E4E3AD07B2465F9">3.8 <span class="Heading">Functions of the truncated variant</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A489A5D79DA9E5C">3.8-1 <span class="Heading">Examples</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CD043E081BF2302">3.8-2 SGrobnerTrunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83C9E598798D5809">3.8-3 CheckHomogeneousNPs</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E33C064875D95CA">3.8-4 BaseQATrunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7C6882DB837A9F5A">3.8-5 DimsQATrunc</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7FBA7F1D79DA883F">3.8-6 FreqsQATrunc</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8706DD3287E82019">3.9 <span class="Heading">Functions of the module variant</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X860966487ED88A43">3.9-1 SGrobnerModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E3160E67C504F37">3.9-2 BaseQM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X813E6A2C8709C9F3">3.9-3 DimQM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X805FB42A7EEF510F">3.9-4 MulQM</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X87D51A8379C50A80">3.9-5 StrongNormalFormNPM</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X79C5DF3782576D98">4 <span class="Heading">Info Level</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7DFB63A97E67C0A1">4.1 <span class="Heading">Introduction</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82D40B0E84383BBC">4.2 <span class="Heading">InfoGBNP</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82D40B0E84383BBC">4.2-1 InfoGBNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8222A2F67E4CC62B">4.2-2 <span class="Heading">What will be printed at level 0</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8552D1FF7EA2B8A6">4.2-3 <span class="Heading">What will be printed at level 1</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CC244E47F903B31">4.2-4 <span class="Heading">What will be printed at level 2</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7FAE244E80397B9A">4.3 <span class="Heading">InfoGBNPTime</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FAE244E80397B9A">4.3-1 InfoGBNPTime</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8222A2F67E4CC62B">4.3-2 <span class="Heading">What will be printed at level 0</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8552D1FF7EA2B8A6">4.3-3 <span class="Heading">What will be printed at level 1</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CC244E47F903B31">4.3-4 <span class="Heading">What will be printed at level 2</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap5.html#X8107DEB279100E13">5 <span class="Heading">NMO Manual</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7DFB63A97E67C0A1">5.1 <span class="Heading">Introduction</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8282EFF97FA1752A">5.2 <span class="Heading">NMO Files within GBNP</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7F83DF528480AEA3">5.3 <span class="Heading">Quickstart</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B44E73581910347">5.3-1 <span class="Heading">NMO Example 1</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X82D4722E7A4DA58B">5.3-2 <span class="Heading">NMO Example 2</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X85A401278794C813">5.3-3 <span class="Heading">NMO Example 3</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C42487D8043F876">5.3-4 <span class="Heading">NMO Example 4</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X86BAEB0C80A24491">5.4 <span class="Heading">Orderings - Internals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X867E06688761CB24">5.4-1 InstallNoncommutativeMonomialOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X804F724282FBA063">5.4-2 IsNoncommutativeMonomialOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7939A8DF8662C60C">5.4-3 LtFunctionListRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E74196084AE9036">5.4-4 NextOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B593F517FF63CDD">5.4-5 ParentAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X850E1F2583F6E2A4">5.4-6 LexicographicTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X82F2AD2583B3CD48">5.4-7 LexicographicIndexTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E1C8F05791E283E">5.4-8 LexicographicPermutation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7EBBF4A07F46E0DD">5.4-9 AuxilliaryTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8228458B86A85279">5.4-10 OrderingLtFunctionListRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X7CDF05BD85AA0EE6">5.5 <span class="Heading">Provided Orderings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X784587377CC4D41F">5.5-1 NCMonomialLeftLengthLexicographicOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7996C01681EC5585">5.5-2 NCMonomialLengthOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7BD70B9C7998C0A7">5.5-3 NCMonomialLeftLexicographicOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7E06DFFA7C4E50C1">5.5-4 NCMonomialCommutativeLexicographicOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B3183F67AEF3C67">5.5-5 NCMonomialWeightOrdering</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X8374E7B780EEE873">5.6 <span class="Heading">Orderings - Externals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7C81894D7A9E9E92">5.6-1 NCLessThanByOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X84BC0A8478272486">5.6-2 NCGreaterThanByOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X817144A57BF6865A">5.6-3 NCEquivalentByOrdering</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X86A2533780F2BC8C">5.6-4 NCSortNP</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X8528D2528613E9A2">5.6-5 <span class="Heading">Flexibility vs. Efficiency</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5.html#X79B90CCE7A05DEEB">5.7 <span class="Heading">Utility Routines</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5.html#X7B758C747AD2344B">5.7-1 <span class="Heading">GBNP Patching Routines</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chapA.html#X7A489A5D79DA9E5C">A <span class="Heading">Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7DFB63A97E67C0A1">A.1 <span class="Heading">Introduction</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X784586E47E2739E3">A.2 <span class="Heading">A simple commutative Gröbner basis computation</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7E1B57AA85C2BA70">A.3 <span class="Heading">A truncated Gröbner basis for Leonard pairs</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X79AC59C482A2E4C1">A.4 <span class="Heading">The truncated variant on two weighted homogeneous polynomials</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7C7742957CEC6E7B">A.5 <span class="Heading">The order of the Weyl group of type E<span class="SimpleMath">_6</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7E39C9738509A036">A.6 <span class="Heading">The gcd of some univariate polynomials</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7F5A6ABA85CDB6E2">A.7 <span class="Heading">From the Tapas book</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7C2CD4FA838EEE64">A.8 <span class="Heading">The Birman-Murakami-Wenzl algebra of type A<span class="SimpleMath">_3</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7B5CA7F379B78CE0">A.9 <span class="Heading">The Birman-Murakami-Wenzl algebra of type A<span class="SimpleMath">_2</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X83C81C987A4DE15F">A.10 <span class="Heading">A commutative example by Mora</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7CAB94A37D580C4A">A.11 <span class="Heading">Tracing an example by Mora</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X8599AE8F7E9E0368">A.12 <span class="Heading"> Finiteness of the Weyl group of type E<span class="SimpleMath">_6</span></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7B1822C67CF83041">A.13 <span class="Heading">Preprocessing for Weyl group computations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7BE4A97886B0930E">A.14 <span class="Heading">A quotient algebra with exponential growth</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X78679D7D80CD8822">A.15 <span class="Heading">A commutative quotient algebra of polynomial growth</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7CE3005580EF632D">A.16 <span class="Heading">An algebra over a finite field</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7E4CEC577A18C8ED">A.17 <span class="Heading">The dihedral group of order 8</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X83328C357FB33D17">A.18 <span class="Heading">The dihedral group of order 8 on another module</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X85DBF3967C4DF5FE">A.19 <span class="Heading">The dihedral group on a non-cyclic module</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X78FCAC347D9D607E">A.20 <span class="Heading">The icosahedral group</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X780C4B777FEA9080">A.21 <span class="Heading">The symmetric inverse monoid for a set of size four</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X84C07DC479FBBCD5">A.22 <span class="Heading">A module of the Hecke algebra of type
A<span class="SimpleMath">_3</span> over GF(3)</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X78C01D1987FEF3FE">A.23 <span class="Heading">Generalized Temperley-Lieb algebras</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X85A9CEF087F3936B">A.24 <span class="Heading">The universal enveloping
algebra of a Lie algebra</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X8498D69D8160E5FF">A.25 <span class="Heading">Serre's exercise</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X8116448A84D69022">A.26 <span class="Heading">Baur and Draisma's transformations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA.html#X7912E411867E5F8B">A.27 <span class="Heading">The cola gene puzzle</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapBib.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd.html"><span class="Heading">Index</span></a></div>
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