/usr/share/gap/doc/ref/chap33.html is in gap-doc 4r8p6-2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 | <?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"
"http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<title>GAP (ref) - Chapter 33: Relations</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<script src="manual.js" type="text/javascript"></script>
<script type="text/javascript">overwriteStyle();</script>
</head>
<body class="chap33" onload="jscontent()">
<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="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap32.html">[Previous Chapter]</a> <a href="chap34.html">[Next Chapter]</a> </div>
<p id="mathjaxlink" class="pcenter"><a href="chap33_mj.html">[MathJax on]</a></p>
<p><a id="X838651287FCCEFD8" name="X838651287FCCEFD8"></a></p>
<div class="ChapSects"><a href="chap33.html#X838651287FCCEFD8">33 <span class="Heading">Relations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7DED7F1F78D31785">33.1 <span class="Heading">General Binary Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X788D722F82165551">33.1-1 IsBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7A1D8EEF8034B0B5">33.1-2 BinaryRelationByElements</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X81878EEF873B34D5">33.1-3 <span class="Heading">IdentityBinaryRelation</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X80DDCDD387BA23F2">33.1-4 EmptyBinaryRelation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7899E59181C46EBB">33.2 <span class="Heading">Properties and Attributes of Binary Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X79D69B667F5FE8FE">33.2-1 IsReflexiveBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X785916A181555368">33.2-2 IsSymmetricBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7823942478124563">33.2-3 IsTransitiveBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X870F72C38550A0A4">33.2-4 IsAntisymmetricBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X782B7C8A8136532F">33.2-5 IsPreOrderBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7A1228207AB4FBA3">33.2-6 IsPartialOrderBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X80D3735C84D1CDC2">33.2-7 IsHasseDiagram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X82D6CB4B7CCE9E25">33.2-8 IsEquivalenceRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X85E2FD8B82652876">33.2-9 Successors</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7B4D22A17E752A91">33.2-10 DegreeOfBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X8278E4457C3C3A0D">33.2-11 PartialOrderOfHasseDiagram</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X78032F927F078E19">33.3 <span class="Heading">Binary Relations on Points</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X79E40E9385274F89">33.3-1 BinaryRelationOnPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7D9323C283867515">33.3-2 RandomBinaryRelationOnPoints</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X8315C7A47CEB6BB3">33.3-3 <span class="Heading">AsBinaryRelationOnPoints</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7D9A14AE799142EF">33.4 <span class="Heading">Closure Operations and Other Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X8252B17C864A4904">33.4-1 ReflexiveClosureBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X820811E9785A7274">33.4-2 SymmetricClosureBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X853BFAD9858DCDF7">33.4-3 TransitiveClosureBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X79672B3A7BCB6991">33.4-4 HasseDiagramBinaryRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X85C22B3D812957C0">33.4-5 StronglyConnectedComponents</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X86AAE6027A3AEF72">33.4-6 PartialOrderByOrderingFunction</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X7DAA67338458BB64">33.5 <span class="Heading">Equivalence Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7A44D73D8150266A">33.5-1 EquivalenceRelationByPartition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X82CD1C00810F6042">33.5-2 EquivalenceRelationByRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7B70215E7E3F9CA4">33.5-3 EquivalenceRelationByPairs</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7C5AA9B97EE42DA5">33.5-4 EquivalenceRelationByProperty</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X85A2A8E27AF52769">33.6 <span class="Heading">Attributes of and Operations on Equivalence Relations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X877389B683DD8F1A">33.6-1 EquivalenceRelationPartition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X79DC914C82D7903B">33.6-2 GeneratorsOfEquivalenceRelationPartition</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X82BE360381476D92">33.6-3 JoinEquivalenceRelations</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap33.html#X79EE13287DEB11B1">33.7 <span class="Heading">Equivalence Classes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X8424996186DB14EA">33.7-1 IsEquivalenceClass</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X78F967E77EB16386">33.7-2 EquivalenceClassRelation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X879439897EF4D728">33.7-3 EquivalenceClasses</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap33.html#X7BB985BA7FD7A82E">33.7-4 EquivalenceClassOfElement</a></span>
</div></div>
</div>
<h3>33 <span class="Heading">Relations</span></h3>
<p>A <em>binary relation</em> <span class="SimpleMath">R</span> on a set <span class="SimpleMath">X</span> is a subset of <span class="SimpleMath">X × X</span>. A binary relation can also be thought of as a (general) mapping from <span class="SimpleMath">X</span> to itself or as a directed graph where each edge represents an element of <span class="SimpleMath">R</span>.</p>
<p>In <strong class="pkg">GAP</strong>, a relation is conceptually represented as a general mapping from <span class="SimpleMath">X</span> to itself. The category <code class="func">IsBinaryRelation</code> (<a href="chap33.html#X788D722F82165551"><span class="RefLink">33.1-1</span></a>) is a synonym for <code class="func">IsEndoGeneralMapping</code> (<a href="chap32.html#X81CFF5F87BBEA8AD"><span class="RefLink">32.13-3</span></a>). Attributes and properties of relations in <strong class="pkg">GAP</strong> are supported for relations, via considering relations as a subset of <span class="SimpleMath">X × X</span>, or as a directed graph; examples include finding the strongly connected components of a relation, via <code class="func">StronglyConnectedComponents</code> (<a href="chap33.html#X85C22B3D812957C0"><span class="RefLink">33.4-5</span></a>), or enumerating the tuples of the relation.</p>
<p><a id="X7DED7F1F78D31785" name="X7DED7F1F78D31785"></a></p>
<h4>33.1 <span class="Heading">General Binary Relations</span></h4>
<p>This section lists general constructors of relations.</p>
<p><a id="X788D722F82165551" name="X788D722F82165551"></a></p>
<h5>33.1-1 IsBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>is exactly the same category as (i.e. a synonym for) <code class="func">IsEndoGeneralMapping</code> (<a href="chap32.html#X81CFF5F87BBEA8AD"><span class="RefLink">32.13-3</span></a>).</p>
<p><a id="X7A1D8EEF8034B0B5" name="X7A1D8EEF8034B0B5"></a></p>
<h5>33.1-2 BinaryRelationByElements</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BinaryRelationByElements</code>( <var class="Arg">domain</var>, <var class="Arg">elms</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the binary relation on <var class="Arg">domain</var> and with underlying relation consisting of the tuples collection <var class="Arg">elms</var>. This construction is similar to <code class="func">GeneralMappingByElements</code> (<a href="chap32.html#X79D0D2F07A14D039"><span class="RefLink">32.2-1</span></a>) where the source and range are the same set.</p>
<p><a id="X81878EEF873B34D5" name="X81878EEF873B34D5"></a></p>
<h5>33.1-3 <span class="Heading">IdentityBinaryRelation</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityBinaryRelation</code>( <var class="Arg">degree</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">‣ IdentityBinaryRelation</code>( <var class="Arg">domain</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the binary relation which consists of diagonal pairs, i.e., pairs of the form <span class="SimpleMath">(x,x)</span>. In the first form if a positive integer <var class="Arg">degree</var> is given then the domain is the set of the integers <span class="SimpleMath">{ 1, ..., <var class="Arg">degree</var> }</span>. In the second form, the objects <span class="SimpleMath">x</span> are from the domain <var class="Arg">domain</var>.</p>
<p><a id="X80DDCDD387BA23F2" name="X80DDCDD387BA23F2"></a></p>
<h5>33.1-4 EmptyBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EmptyBinaryRelation</code>( <var class="Arg">degree</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">‣ EmptyBinaryRelation</code>( <var class="Arg">domain</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>is the relation with <var class="Arg">R</var> empty. In the first form of the command with <var class="Arg">degree</var> an integer, the domain is the set of points <span class="SimpleMath">{ 1, ..., <var class="Arg">degree</var> }</span>. In the second form, the domain is that given by the argument <var class="Arg">domain</var>.</p>
<p><a id="X7899E59181C46EBB" name="X7899E59181C46EBB"></a></p>
<h4>33.2 <span class="Heading">Properties and Attributes of Binary Relations</span></h4>
<p><a id="X79D69B667F5FE8FE" name="X79D69B667F5FE8FE"></a></p>
<h5>33.2-1 IsReflexiveBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsReflexiveBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">R</var> is reflexive, and <code class="keyw">false</code> otherwise.</p>
<p>A binary relation <span class="SimpleMath">R</span> (as a set of pairs) on a set <span class="SimpleMath">X</span> is <em>reflexive</em> if for all <span class="SimpleMath">x ∈ X</span>, <span class="SimpleMath">(x,x) ∈ R</span>. Alternatively, <span class="SimpleMath">R</span> as a mapping is reflexive if for all <span class="SimpleMath">x ∈ X</span>, <span class="SimpleMath">x</span> is an element of the image set <span class="SimpleMath">R(x)</span>.</p>
<p>A reflexive binary relation is necessarily a total endomorphic mapping (tested via <code class="func">IsTotal</code> (<a href="chap32.html#X83C7494E828CC9C8"><span class="RefLink">32.3-1</span></a>)).</p>
<p><a id="X785916A181555368" name="X785916A181555368"></a></p>
<h5>33.2-2 IsSymmetricBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsSymmetricBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">R</var> is symmetric, and <code class="keyw">false</code> otherwise.</p>
<p>A binary relation <span class="SimpleMath">R</span> (as a set of pairs) on a set <span class="SimpleMath">X</span> is <em>symmetric</em> if <span class="SimpleMath">(x,y) ∈ R</span> then <span class="SimpleMath">(y,x) ∈ R</span>. Alternatively, <span class="SimpleMath">R</span> as a mapping is symmetric if for all <span class="SimpleMath">x ∈ X</span>, the preimage set of <span class="SimpleMath">x</span> under <span class="SimpleMath">R</span> equals the image set <span class="SimpleMath">R(x)</span>.</p>
<p><a id="X7823942478124563" name="X7823942478124563"></a></p>
<h5>33.2-3 IsTransitiveBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsTransitiveBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">R</var> is transitive, and <code class="keyw">false</code> otherwise.</p>
<p>A binary relation <var class="Arg">R</var> (as a set of pairs) on a set <span class="SimpleMath">X</span> is <em>transitive</em> if <span class="SimpleMath">(x,y), (y,z) ∈ R</span> implies <span class="SimpleMath">(x,z) ∈ R</span>. Alternatively, <span class="SimpleMath">R</span> as a mapping is transitive if for all <span class="SimpleMath">x ∈ X</span>, the image set <span class="SimpleMath">R(R(x))</span> of the image set <span class="SimpleMath">R(x)</span> of <span class="SimpleMath">x</span> is a subset of <span class="SimpleMath">R(x)</span>.</p>
<p><a id="X870F72C38550A0A4" name="X870F72C38550A0A4"></a></p>
<h5>33.2-4 IsAntisymmetricBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAntisymmetricBinaryRelation</code>( <var class="Arg">rel</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">rel</var> is antisymmetric, and <code class="keyw">false</code> otherwise.</p>
<p>A binary relation <var class="Arg">R</var> (as a set of pairs) on a set <span class="SimpleMath">X</span> is <em>antisymmetric</em> if <span class="SimpleMath">(x,y), (y,x) ∈ R</span> implies <span class="SimpleMath">x = y</span>. Alternatively, <span class="SimpleMath">R</span> as a mapping is antisymmetric if for all <span class="SimpleMath">x ∈ X</span>, the intersection of the preimage set of <span class="SimpleMath">x</span> under <span class="SimpleMath">R</span> and the image set <span class="SimpleMath">R(x)</span> is <span class="SimpleMath">{ x }</span>.</p>
<p><a id="X782B7C8A8136532F" name="X782B7C8A8136532F"></a></p>
<h5>33.2-5 IsPreOrderBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPreOrderBinaryRelation</code>( <var class="Arg">rel</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">rel</var> is a preorder, and <code class="keyw">false</code> otherwise.</p>
<p>A <em>preorder</em> is a binary relation that is both reflexive and transitive.</p>
<p><a id="X7A1228207AB4FBA3" name="X7A1228207AB4FBA3"></a></p>
<h5>33.2-6 IsPartialOrderBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsPartialOrderBinaryRelation</code>( <var class="Arg">rel</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">rel</var> is a partial order, and <code class="keyw">false</code> otherwise.</p>
<p>A <em>partial order</em> is a preorder which is also antisymmetric.</p>
<p><a id="X80D3735C84D1CDC2" name="X80D3735C84D1CDC2"></a></p>
<h5>33.2-7 IsHasseDiagram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsHasseDiagram</code>( <var class="Arg">rel</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">rel</var> is a Hasse Diagram of a partial order, i.e., was computed via <code class="func">HasseDiagramBinaryRelation</code> (<a href="chap33.html#X79672B3A7BCB6991"><span class="RefLink">33.4-4</span></a>).</p>
<p><a id="X82D6CB4B7CCE9E25" name="X82D6CB4B7CCE9E25"></a></p>
<h5>33.2-8 IsEquivalenceRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEquivalenceRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the binary relation <var class="Arg">R</var> is an equivalence relation, and <code class="keyw">false</code> otherwise.</p>
<p>Recall, that a relation <var class="Arg">R</var> is an <em>equivalence relation</em> if it is symmetric, transitive, and reflexive.</p>
<p><a id="X85E2FD8B82652876" name="X85E2FD8B82652876"></a></p>
<h5>33.2-9 Successors</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Successors</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the list of images of a binary relation <var class="Arg">R</var>. If the underlying domain of the relation is not <span class="SimpleMath">{ 1, ..., n }</span>, for some positive integer <span class="SimpleMath">n</span>, then an error is signalled.</p>
<p>The returned value of <code class="func">Successors</code> is a list of lists where the lists are ordered as the elements according to the sorted order of the underlying set of <var class="Arg">R</var>. Each list consists of the images of the element whose index is the same as the list with the underlying set in sorted order.</p>
<p>The <code class="func">Successors</code> of a relation is the adjacency list representation of the relation.</p>
<p><a id="X7B4D22A17E752A91" name="X7B4D22A17E752A91"></a></p>
<h5>33.2-10 DegreeOfBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DegreeOfBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the size of the underlying domain of the binary relation <var class="Arg">R</var>. This is most natural when working with a binary relation on points.</p>
<p><a id="X8278E4457C3C3A0D" name="X8278E4457C3C3A0D"></a></p>
<h5>33.2-11 PartialOrderOfHasseDiagram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialOrderOfHasseDiagram</code>( <var class="Arg">HD</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is the partial order associated with the Hasse Diagram <var class="Arg">HD</var> i.e. the partial order generated by the reflexive and transitive closure of <var class="Arg">HD</var>.</p>
<p><a id="X78032F927F078E19" name="X78032F927F078E19"></a></p>
<h4>33.3 <span class="Heading">Binary Relations on Points</span></h4>
<p>We have special construction methods when the underlying <var class="Arg">X</var> of our relation is the set of integers <span class="SimpleMath">{ 1, ..., n }</span>.</p>
<p><a id="X79E40E9385274F89" name="X79E40E9385274F89"></a></p>
<h5>33.3-1 BinaryRelationOnPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BinaryRelationOnPoints</code>( <var class="Arg">list</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">‣ BinaryRelationOnPointsNC</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Given a list of <span class="SimpleMath">n</span> lists, each containing elements from the set <span class="SimpleMath">{ 1, ..., n }</span>, this function constructs a binary relation such that <span class="SimpleMath">1</span> is related to <var class="Arg">list</var><code class="code">[1]</code>, <span class="SimpleMath">2</span> to <var class="Arg">list</var><code class="code">[2]</code> and so on. The first version checks whether the list supplied is valid. The the <code class="code">NC</code> version skips this check.</p>
<p><a id="X7D9323C283867515" name="X7D9323C283867515"></a></p>
<h5>33.3-2 RandomBinaryRelationOnPoints</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomBinaryRelationOnPoints</code>( <var class="Arg">degree</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates a relation on points with degree <var class="Arg">degree</var>.</p>
<p><a id="X8315C7A47CEB6BB3" name="X8315C7A47CEB6BB3"></a></p>
<h5>33.3-3 <span class="Heading">AsBinaryRelationOnPoints</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AsBinaryRelationOnPoints</code>( <var class="Arg">trans</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">‣ AsBinaryRelationOnPoints</code>( <var class="Arg">perm</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">‣ AsBinaryRelationOnPoints</code>( <var class="Arg">rel</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>return the relation on points represented by general relation <var class="Arg">rel</var>, transformation <var class="Arg">trans</var> or permutation <var class="Arg">perm</var>. If <var class="Arg">rel</var> is already a binary relation on points then <var class="Arg">rel</var> is returned.</p>
<p>Transformations and permutations are special general endomorphic mappings and have a natural representation as a binary relation on points.</p>
<p>In the last form, an isomorphic relation on points is constructed where the points are indices of the elements of the underlying domain in sorted order.</p>
<p><a id="X7D9A14AE799142EF" name="X7D9A14AE799142EF"></a></p>
<h4>33.4 <span class="Heading">Closure Operations and Other Constructors</span></h4>
<p><a id="X8252B17C864A4904" name="X8252B17C864A4904"></a></p>
<h5>33.4-1 ReflexiveClosureBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReflexiveClosureBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the smallest binary relation containing the binary relation <var class="Arg">R</var> which is reflexive. This closure inherits the properties symmetric and transitive from <var class="Arg">R</var>. E.g., if <var class="Arg">R</var> is symmetric then its reflexive closure is also.</p>
<p><a id="X820811E9785A7274" name="X820811E9785A7274"></a></p>
<h5>33.4-2 SymmetricClosureBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SymmetricClosureBinaryRelation</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the smallest binary relation containing the binary relation <var class="Arg">R</var> which is symmetric. This closure inherits the properties reflexive and transitive from <var class="Arg">R</var>. E.g., if <var class="Arg">R</var> is reflexive then its symmetric closure is also.</p>
<p><a id="X853BFAD9858DCDF7" name="X853BFAD9858DCDF7"></a></p>
<h5>33.4-3 TransitiveClosureBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransitiveClosureBinaryRelation</code>( <var class="Arg">rel</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the smallest binary relation containing the binary relation <var class="Arg">R</var> which is transitive. This closure inherits the properties reflexive and symmetric from <var class="Arg">R</var>. E.g., if <var class="Arg">R</var> is symmetric then its transitive closure is also.</p>
<p><code class="func">TransitiveClosureBinaryRelation</code> is a modified version of the Floyd-Warshall method of solving the all-pairs shortest-paths problem on a directed graph. Its asymptotic runtime is <span class="SimpleMath">O(n^3)</span> where <span class="SimpleMath">n</span> is the size of the vertex set. It only assumes there is an arbitrary (but fixed) ordering of the vertex set.</p>
<p><a id="X79672B3A7BCB6991" name="X79672B3A7BCB6991"></a></p>
<h5>33.4-4 HasseDiagramBinaryRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HasseDiagramBinaryRelation</code>( <var class="Arg">partial-order</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>is the smallest relation contained in the partial order <var class="Arg">partial-order</var> whose reflexive and transitive closure is equal to <var class="Arg">partial-order</var>.</p>
<p><a id="X85C22B3D812957C0" name="X85C22B3D812957C0"></a></p>
<h5>33.4-5 StronglyConnectedComponents</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StronglyConnectedComponents</code>( <var class="Arg">R</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>returns an equivalence relation on the vertices of the binary relation <var class="Arg">R</var>.</p>
<p><a id="X86AAE6027A3AEF72" name="X86AAE6027A3AEF72"></a></p>
<h5>33.4-6 PartialOrderByOrderingFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PartialOrderByOrderingFunction</code>( <var class="Arg">dom</var>, <var class="Arg">orderfunc</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs a partial order whose elements are from the domain <var class="Arg">dom</var> and are ordered using the ordering function <var class="Arg">orderfunc</var>. The ordering function must be a binary function returning a boolean value. If the ordering function does not describe a partial order then <code class="keyw">fail</code> is returned.</p>
<p><a id="X7DAA67338458BB64" name="X7DAA67338458BB64"></a></p>
<h4>33.5 <span class="Heading">Equivalence Relations</span></h4>
<p>An <em>equivalence relation</em> <var class="Arg">E</var> over the set <var class="Arg">X</var> is a relation on <var class="Arg">X</var> which is reflexive, symmetric, and transitive. A <em>partition</em> <var class="Arg">P</var> is a set of subsets of <var class="Arg">X</var> such that for all <span class="SimpleMath">R, S ∈ P</span>, <span class="SimpleMath">R ∩ S</span> is the empty set and <span class="SimpleMath">∪ P = X</span>. An equivalence relation induces a partition such that if <span class="SimpleMath">(x,y) ∈ E</span> then <span class="SimpleMath">x, y</span> are in the same element of <var class="Arg">P</var>.</p>
<p>Like all binary relations in <strong class="pkg">GAP</strong> equivalence relations are regarded as general endomorphic mappings (and the operations, properties and attributes of general mappings are available). However, partitions provide an efficient way of representing equivalence relations. Moreover, only the non-singleton classes or blocks are listed allowing for small equivalence relations to be represented on infinite sets. Hence the main attribute of equivalence relations is <code class="func">EquivalenceRelationPartition</code> (<a href="chap33.html#X877389B683DD8F1A"><span class="RefLink">33.6-1</span></a>) which provides the partition induced by the given equivalence.</p>
<p><a id="X7A44D73D8150266A" name="X7A44D73D8150266A"></a></p>
<h5>33.5-1 EquivalenceRelationByPartition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceRelationByPartition</code>( <var class="Arg">domain</var>, <var class="Arg">list</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">‣ EquivalenceRelationByPartitionNC</code>( <var class="Arg">domain</var>, <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>constructs the equivalence relation over the set <var class="Arg">domain</var> which induces the partition represented by <var class="Arg">list</var>. This representation includes only the non-trivial blocks (or equivalent classes). <var class="Arg">list</var> is a list of lists, each of these lists contain elements of <var class="Arg">domain</var> and are pairwise mutually exclusive.</p>
<p>The list of lists do not need to be in any order nor do the elements in the blocks (see <code class="func">EquivalenceRelationPartition</code> (<a href="chap33.html#X877389B683DD8F1A"><span class="RefLink">33.6-1</span></a>)). a list of elements of <var class="Arg">domain</var> The partition <var class="Arg">list</var> is a list of lists, each of these is a list of elements of <var class="Arg">domain</var> that makes up a block (or equivalent class). The <var class="Arg">domain</var> is the domain over which the relation is defined, and <var class="Arg">list</var> is a list of lists, each of these is a list of elements of <var class="Arg">domain</var> which are related to each other. <var class="Arg">list</var> need only contain the nontrivial blocks and singletons will be ignored. The <code class="code">NC</code> version will not check to see if the lists are pairwise mutually exclusive or that they contain only elements of the domain.</p>
<p><a id="X82CD1C00810F6042" name="X82CD1C00810F6042"></a></p>
<h5>33.5-2 EquivalenceRelationByRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceRelationByRelation</code>( <var class="Arg">rel</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the smallest equivalence relation containing the binary relation <var class="Arg">rel</var>.</p>
<p><a id="X7B70215E7E3F9CA4" name="X7B70215E7E3F9CA4"></a></p>
<h5>33.5-3 EquivalenceRelationByPairs</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceRelationByPairs</code>( <var class="Arg">D</var>, <var class="Arg">elms</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">‣ EquivalenceRelationByPairsNC</code>( <var class="Arg">D</var>, <var class="Arg">elms</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>return the smallest equivalence relation on the domain <var class="Arg">D</var> such that every pair in <var class="Arg">elms</var> is in the relation.</p>
<p>In the <code class="code">NC</code> form, it is not checked that <var class="Arg">elms</var> are in the domain <var class="Arg">D</var>.</p>
<p><a id="X7C5AA9B97EE42DA5" name="X7C5AA9B97EE42DA5"></a></p>
<h5>33.5-4 EquivalenceRelationByProperty</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceRelationByProperty</code>( <var class="Arg">domain</var>, <var class="Arg">property</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>creates an equivalence relation on <var class="Arg">domain</var> whose only defining datum is that of having the property <var class="Arg">property</var>.</p>
<p><a id="X85A2A8E27AF52769" name="X85A2A8E27AF52769"></a></p>
<h4>33.6 <span class="Heading">Attributes of and Operations on Equivalence Relations</span></h4>
<p><a id="X877389B683DD8F1A" name="X877389B683DD8F1A"></a></p>
<h5>33.6-1 EquivalenceRelationPartition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceRelationPartition</code>( <var class="Arg">equiv</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of lists of elements of the underlying set of the equivalence relation <var class="Arg">equiv</var>. The lists are precisely the nonsingleton equivalence classes of the equivalence. This allows us to describe "small" equivalences on infinite sets.</p>
<p><a id="X79DC914C82D7903B" name="X79DC914C82D7903B"></a></p>
<h5>33.6-2 GeneratorsOfEquivalenceRelationPartition</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GeneratorsOfEquivalenceRelationPartition</code>( <var class="Arg">equiv</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>is a set of generating pairs for the equivalence relation <var class="Arg">equiv</var>. This set is not unique. The equivalence <var class="Arg">equiv</var> is the smallest equivalence relation over the underlying set which contains the generating pairs.</p>
<p><a id="X82BE360381476D92" name="X82BE360381476D92"></a></p>
<h5>33.6-3 JoinEquivalenceRelations</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ JoinEquivalenceRelations</code>( <var class="Arg">equiv1</var>, <var class="Arg">equiv2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MeetEquivalenceRelations</code>( <var class="Arg">equiv1</var>, <var class="Arg">equiv2</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><code class="func">JoinEquivalenceRelations</code> returns the smallest equivalence relation containing both the equivalence relations <var class="Arg">equiv1</var> and <var class="Arg">equiv2</var>.</p>
<p><code class="func">MeetEquivalenceRelations</code> returns the intersection of the two equivalence relations <var class="Arg">equiv1</var> and <var class="Arg">equiv2</var>.</p>
<p><a id="X79EE13287DEB11B1" name="X79EE13287DEB11B1"></a></p>
<h4>33.7 <span class="Heading">Equivalence Classes</span></h4>
<p><a id="X8424996186DB14EA" name="X8424996186DB14EA"></a></p>
<h5>33.7-1 IsEquivalenceClass</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsEquivalenceClass</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>returns <code class="keyw">true</code> if the object <var class="Arg">obj</var> is an equivalence class, and <code class="keyw">false</code> otherwise.</p>
<p>An <em>equivalence class</em> is a collection of elements which are mutually related to each other in the associated equivalence relation. Note, this is a special category of objects and not just a list of elements.</p>
<p><a id="X78F967E77EB16386" name="X78F967E77EB16386"></a></p>
<h5>33.7-2 EquivalenceClassRelation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceClassRelation</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the equivalence relation of which <var class="Arg">C</var> is a class.</p>
<p><a id="X879439897EF4D728" name="X879439897EF4D728"></a></p>
<h5>33.7-3 EquivalenceClasses</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceClasses</code>( <var class="Arg">rel</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a list of all equivalence classes of the equivalence relation <var class="Arg">rel</var>. Note that it is possible for different methods to yield the list in different orders, so that for two equivalence relations <span class="SimpleMath">c1</span> and <span class="SimpleMath">c2</span> we may have <span class="SimpleMath">c1 = c2</span> without having <code class="code">EquivalenceClasses</code><span class="SimpleMath">( c1 ) =</span><code class="code">EquivalenceClasses</code><span class="SimpleMath">( c2 )</span>.</p>
<p><a id="X7BB985BA7FD7A82E" name="X7BB985BA7FD7A82E"></a></p>
<h5>33.7-4 EquivalenceClassOfElement</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceClassOfElement</code>( <var class="Arg">rel</var>, <var class="Arg">elt</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EquivalenceClassOfElementNC</code>( <var class="Arg">rel</var>, <var class="Arg">elt</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>return the equivalence class of <var class="Arg">elt</var> in the binary relation <var class="Arg">rel</var>, where <var class="Arg">elt</var> is an element (i.e. a pair) of the domain of <var class="Arg">rel</var>. In the <code class="code">NC</code> form, it is not checked that <var class="Arg">elt</var> is in the domain over which <var class="Arg">rel</var> is defined.</p>
<div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#contents">[Contents]</a> <a href="chap32.html">[Previous Chapter]</a> <a href="chap34.html">[Next Chapter]</a> </div>
<div class="chlinkbot"><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="chap6.html">6</a> <a href="chap7.html">7</a> <a href="chap8.html">8</a> <a href="chap9.html">9</a> <a href="chap10.html">10</a> <a href="chap11.html">11</a> <a href="chap12.html">12</a> <a href="chap13.html">13</a> <a href="chap14.html">14</a> <a href="chap15.html">15</a> <a href="chap16.html">16</a> <a href="chap17.html">17</a> <a href="chap18.html">18</a> <a href="chap19.html">19</a> <a href="chap20.html">20</a> <a href="chap21.html">21</a> <a href="chap22.html">22</a> <a href="chap23.html">23</a> <a href="chap24.html">24</a> <a href="chap25.html">25</a> <a href="chap26.html">26</a> <a href="chap27.html">27</a> <a href="chap28.html">28</a> <a href="chap29.html">29</a> <a href="chap30.html">30</a> <a href="chap31.html">31</a> <a href="chap32.html">32</a> <a href="chap33.html">33</a> <a href="chap34.html">34</a> <a href="chap35.html">35</a> <a href="chap36.html">36</a> <a href="chap37.html">37</a> <a href="chap38.html">38</a> <a href="chap39.html">39</a> <a href="chap40.html">40</a> <a href="chap41.html">41</a> <a href="chap42.html">42</a> <a href="chap43.html">43</a> <a href="chap44.html">44</a> <a href="chap45.html">45</a> <a href="chap46.html">46</a> <a href="chap47.html">47</a> <a href="chap48.html">48</a> <a href="chap49.html">49</a> <a href="chap50.html">50</a> <a href="chap51.html">51</a> <a href="chap52.html">52</a> <a href="chap53.html">53</a> <a href="chap54.html">54</a> <a href="chap55.html">55</a> <a href="chap56.html">56</a> <a href="chap57.html">57</a> <a href="chap58.html">58</a> <a href="chap59.html">59</a> <a href="chap60.html">60</a> <a href="chap61.html">61</a> <a href="chap62.html">62</a> <a href="chap63.html">63</a> <a href="chap64.html">64</a> <a href="chap65.html">65</a> <a href="chap66.html">66</a> <a href="chap67.html">67</a> <a href="chap68.html">68</a> <a href="chap69.html">69</a> <a href="chap70.html">70</a> <a href="chap71.html">71</a> <a href="chap72.html">72</a> <a href="chap73.html">73</a> <a href="chap74.html">74</a> <a href="chap75.html">75</a> <a href="chap76.html">76</a> <a href="chap77.html">77</a> <a href="chap78.html">78</a> <a href="chap79.html">79</a> <a href="chap80.html">80</a> <a href="chap81.html">81</a> <a href="chap82.html">82</a> <a href="chap83.html">83</a> <a href="chap84.html">84</a> <a href="chap85.html">85</a> <a href="chap86.html">86</a> <a href="chap87.html">87</a> <a href="chapBib.html">Bib</a> <a href="chapInd.html">Ind</a> </div>
<hr />
<p class="foot">generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
</body>
</html>
|