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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="chapBib.html">Bib</a>  <a href="chapInd.html">Ind</a>  </div>

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<h1>Smallsemi</h1>


<h2>A library of small semigroups</h2>

<p>
    Version 0.7.2</p>

<p>
    25 February 2025
  </p>

</div>
<p><b>
    Andreas Distler


  </b>
<br />Email: <span class="URL"><a href="mailto:a.distler@tu-bs.de">a.distler@tu-bs.de</a></span>
</p><p><b>
    James Mitchell




  </b>
<br />Email: <span class="URL"><a href="mailto:jdm3@st-andrews.ac.uk">jdm3@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://jdbm.me">https://jdbm.me</a></span>
<br />Address: <br />Mathematical Institute<br /> North Haugh<br /> St Andrews<br /> Fife<br /> KY16 9SS<br /> Scotland<br />
</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2008-24 A. Distler & J. D. Mitchell.</p>

<p><strong class="pkg">Smallsemi</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 3 of the license, or (at your option) any later version.</p>

<p><strong class="pkg">Smallsemi</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>A copy of the GNU General Public License is available in the file 'LICENSE'; for the latest version see <span class="URL"><a href="https://www.gnu.org/licenses/#GPL">https://www.gnu.org/licenses/#GPL</a></span>.</p>

<p>This file is part of <strong class="pkg">Smallsemi</strong>, though as documentation it is released under the GNU Free Documentation License (see <span class="URL"><a href="https://www.gnu.org/licenses/#FDL">https://www.gnu.org/licenses/#FDL</a></span>).</p>

<p><a id="X82A988D47DFAFCFA" name="X82A988D47DFAFCFA"></a></p>
<h3>Acknowledgements</h3>
<p>We would like to thank Tom Kelsey for making this library possible by running all the initial computations in Minion <a href="chapBib.html#biBminion">[GJM06]</a>. <br /> The first author acknowledges financial support of the University of St Andrews. The second author acknowledges support of EPSRC grant number GR/S/56085/01.</p>

<p><a id="X7982162280BC7A61" name="X7982162280BC7A61"></a></p>
<h3>Colophon</h3>
<p>If you find any bugs or have any suggestions or comments, we would very much appreciate it if you would let us know via our issue tracker (see <span class="URL"><a href="https://github.com/gap-packages/smallsemi/issues">https://github.com/gap-packages/smallsemi/issues</a></span>).</p>

<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#X85A08CF187A6D986">1.1 <span class="Heading">Requirements</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X82C5588B7E8B2725">1.1-1 <span class="Heading">Operating System</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X78F27CCA81555251">1.1-2 <span class="Heading">RAM</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87E4E0377BF1EAC6">1.1-3 <span class="Heading">Disk Space</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X796A04C37CB7AC04">1.2 <span class="Heading">Installation and Setup</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C87C34C79B9D28D">1.2-1 <span class="Heading">Download and Extract <strong class="pkg">Smallsemi</strong></span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86BA319E87365B5C">1.2-2 <span class="Heading">Contents</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X861ED1338181C66D">1.2-3 <span class="Heading">Loading</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X803939AE7C3565DA">1.2-4 <span class="Heading">Memory Issues</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DE7E7187BE24368">1.2-5 <span class="Heading">Testing</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7B3500B984306465">1.2-6 <span class="Heading">Customizing</span></a>
</span>
</div></div>
</div>
<div class="ContChap"><a href="chap2.html#X7E933CC381DA3D7C">2 <span class="Heading">The Data in the Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X7A8F62A38080168D">2.1 <span class="Heading">Creation of the Semigroups</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X79038EE8873A1B4B">2.2 <span class="Heading">Storing the Semigroups</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X7CDC63A27F7790AA">3 <span class="Heading">Extended Examples</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X80DFE0117B7A7C24">3.1 <span class="Heading">Lists, Enumerators and Iterators of Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X790CB8D686E0A336">3.1-1 <span class="Heading">Precomputed properties</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X804593297D7EEF68">3.1-2 <span class="Heading">User functions</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7E4F0756878EB958">3.1-3 <span class="Heading">Semigroups of order 8</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8772952F7B2CFDE1">3.2 <span class="Heading">Identifying Semigroups</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap4.html#X87F1120883F5B4D0">4 <span class="Heading">Functionality</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X83F5AF81859D6CBF">4.1 <span class="Heading">Individual Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8538248D78185960">4.1-1 SmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X857428A57D3FFACB">4.1-2 IsSmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EC241FB7985775E">4.1-3 IsSmallSemigroupElt</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X793FF84E80B334D1">4.1-4 RecoverMultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X813727FD851302B7">4.1-5 SemigroupByMultiplicationTableNC</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X788211A07D67C282">4.1-6 IdSmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79D38A3886C7431D">4.1-7 EquivalenceSmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8078FC78871FA496">4.1-8 InfoSmallsemi</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X841BB74986A73272">4.1-9 UnloadSmallsemiData</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X78274024827F306D">4.2 <span class="Heading">Properties of Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86A4F95783A18702">4.2-1 Annihilators</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E1195927C454C8A">4.2-2 DiagonalOfMultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7DCF7C368665E94D">4.2-3 DisplaySmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86B662117F4C4C7F">4.2-4 IndexPeriod</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C8DB14587D1B55A">4.2-5 IsBand</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EFDBA687DCDA6FA">4.2-6 IsBrandtSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81DE11987BB81017">4.2-7 IsCliffordSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X843EFDA4807FDC31">4.2-8 IsCommutativeSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AFA23AF819FBF3D">4.2-9 IsCompletelyRegularSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D51FD2B7AA3E8DF">4.2-10 IsFullTransformationSemigroupCopy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X852F29E8795FA489">4.2-11 IsGroupAsSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X835484C481CF3DDD">4.2-12 <span class="Heading">IsIdempotentGenerated</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83F1529479D56665">4.2-13 IsInverseSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E9261367C8C52C0">4.2-14 IsLeftZeroSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79D46BAB7E327AD1">4.2-15 IsMonogenicSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E4DEECD7CD9886D">4.2-16 IsMonoidAsSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79FF88207AFC8330">4.2-17 IsMultSemigroupOfNearRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83D4ED237BB929AF">4.2-18 IsNGeneratedSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87BF19367BA2B9ED">4.2-19 IsNIdempotentSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X780ADE31828F0848">4.2-20 <span class="Heading">IsNilpotentSemigroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7935C476808C8773">4.2-21 IsOrthodoxSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E9B674D781B072C">4.2-22 IsRectangularBand</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C4663827C5ACEF1">4.2-23 IsRegularSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CB099958658F979">4.2-24 IsRightZeroSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X846FC6247EE31607">4.2-25 IsSelfDualSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D3ACD8A7C0AB34C">4.2-26 IsSemigroupWithClosedIdempotents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82D229727C6278EC">4.2-27 IsSemigroupWithZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X836F4692839F4874">4.2-28 <span class="Heading">IsSimpleSemigroup</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85A3F78C864507EB">4.2-29 IsSingularSemigroupCopy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85F7E5CD86F0643B">4.2-30 IsZeroGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81A1882181B75CC9">4.2-31 IsZeroSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8193A60F839C064E">4.2-32 IsZeroSimpleSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81D15723804771E2">4.2-33 MinimalGeneratingSet</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D1C336E7B0A059C">4.2-34 NilpotencyDegree</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X866B154E7F54AE62">4.3 <span class="Heading">Nilpotent semigroups by coclass</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X819BD88C78976AFD">4.3-1 NilpotentSemigroupsByCoclass</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7D6E81E37E3AAED8">4.4 <span class="Heading">Starred Green's relations
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84EAD3BC80389059">4.4-1 IsStarRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A8059EC848E0A77">4.4-2 RStarRelation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D30CDC386C8816A">4.4-3 RStarClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83E1C16B8247ECBD">4.4-4 RStarClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82A391177D13A7E0">4.4-5 IsStarClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83B9C0417C03A220">4.4-6 RStarClasses</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X82F9C36C86006857">4.5 <span class="Heading">Families of Semigroups</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81DC0FE28043C9B0">4.5-1 AllSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X831E543E83DDFDEA">4.5-2 EnumeratorOfSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F8EF88E7EB417A8">4.5-3 EnumeratorOfSmallSemigroupsByIds</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A14F5A58020DCFF">4.5-4 ArgumentsUsedToCreate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X814BA9F778B06D47">4.5-5 IdsOfSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7BF0DCA5853A5C22">4.5-6 IsEnumeratorOfSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83C27CD97895698A">4.5-7 IsIdSmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X803366027F328BC9">4.5-8 IsIteratorOfSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D6BFDE17A9BEEC3">4.5-9 IteratorOfSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X867F3E967CF3AFE9">4.5-10 Nr3NilpotentSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8257C9207DDF2451">4.5-11 NrSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7C71885F7DF31EC4">4.5-12 OneSmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EACCD1F7DF9DD3C">4.5-13 PositionsOfSmallSemigroupsIn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FAF09BF7CC1C265">4.5-14 PositionsOfSmallSemigroups</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79AFE7FC81C0DF37">4.5-15 PrecomputedSmallSemisInfo</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A63356C85E45F6F">4.5-16 RandomSmallSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X83C6631B7D53AE02">4.5-17 SizesOfSmallSemigroupsIn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X87C50557821D4786">4.5-18 UpToIsomorphism</a></span>
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