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<h1><strong class="pkg">Circle</strong></h1>


<h2>Adjoint groups of finite rings</h2>

<p>Version 1.6.6</p>

<p>25 February 2023</p>

</div>
<p><b>Olexandr Konovalov
          
       
           
  </b>
<br />Email: <span class="URL"><a href="mailto:obk1@st-andrews.ac.uk">obk1@st-andrews.ac.uk</a></span>
<br />Homepage: <span class="URL"><a href="https://alex-konovalov.github.io/">https://alex-konovalov.github.io/</a></span>
<br />Address: <br />School of Computer Science<br /> University of St Andrews<br /> Jack Cole Building, North Haugh,<br /> St Andrews, Fife, KY16 9SX, Scotland
</p><p><b>Panagiotis Soules
          
                  
  </b>
<br />Email: <span class="URL"><a href="mailto:psoules@math.uoa.gr">psoules@math.uoa.gr</a></span>
<br />Address: <br />Department of Mathematics<br /> National and Capodistrian University of Athens<br /> Panepistimioupolis, GR-15784, Athens, Greece
</p>

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<h3>Abstract</h3>
<p>The <strong class="pkg">GAP</strong>4 package <strong class="pkg">Circle</strong> extends the <strong class="pkg">GAP</strong> functionality for computations in adjoint groups of associative rings. It provides functionality to construct circle objects that will respect the circle multiplication <span class="SimpleMath">r ⋅ s = r + s + rs</span>, and to compute adjoint semigroups and adjoint groups of finite rings. Also it may serve as an example of extending the <strong class="pkg">GAP</strong> system with new multiplicative objects.</p>

<p><a id="X81488B807F2A1CF1" name="X81488B807F2A1CF1"></a></p>
<h3>Copyright</h3>
<p>© 2006-2023 by Olexandr Konovalov and Panagiotis Soules</p>

<p><strong class="pkg">Circle</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. For details, see the FSF's own site https://www.gnu.org/licenses/gpl.html.



<p>If you obtained <strong class="pkg">Circle</strong>, we would be grateful for a short notification sent to one of the authors.</p>

<p>If you publish a result which was partially obtained with the usage of <strong class="pkg">Circle</strong>, please cite it in the following form:</p>

<p>O. Konovalov, P. Soules. <em>Circle --- Adjoint groups of finite rings, Version 1.6.6;</em> 2023 (<span class="URL"><a href="https://gap-packages.github.io/circle/">https://gap-packages.github.io/circle/</a></span>).</p>

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<h3>Acknowledgements</h3>
<p>We acknowledge very much Alexander Hulpke and James Mitchell for their helpful comments and advices, and the referee for testing the package and useful suggestions.</p>

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<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#X8557083378F2A3B2">1.1 <span class="Heading">General aims</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7DB566D5785B7DBC">1.2 <span class="Heading">Installation and system requirements</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2.html#X8404D6997A466953">2 <span class="Heading">Implementing circle objects</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X86492955868108EC">2.1 <span class="Heading">First attempts</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X852C1F3281137DD6">2.2 <span class="Heading">Defining circle objects</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.html#X85B1413E7FBC8ACB">2.3 <span class="Heading">Installing operations for circle objects</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap3.html#X81CE94FA8343B1D8">3 <span class="Heading"><strong class="pkg">Circle</strong> functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7E25EF1786B5F87C">3.1 <span class="Heading">Circle objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F5C18AA7B433BDD">3.1-1 CircleObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X874B2B2A7F5A9A78">3.1-2 UnderlyingRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X810FB77F860F888D">3.1-3 IsCircleObject</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8469C7F67A8864B3">3.1-4 IsPositionalObjectOneSlotRep</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7F88A0017DC2E878">3.1-5 CircleFamily</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X85ECB0B482AB3170">3.2 <span class="Heading">Operations with circle objects</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X8129A6877FFD804B">3.2-1 One</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X82EC4F49877D6EB1">3.2-2 InverseOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X85CBFBAE78DE72E8">3.2-3 IsUnit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7CE7866F7CD00709">3.2-4 IsCircleUnit</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X86F80DD8823966F7">3.3 <span class="Heading">Construction of the adjoint semigroup and adjoint group</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X83993FD0848D0C80">3.3-1 AdjointSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X868FAC7E87D11137">3.3-2 AdjointGroup</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X80DFB24F8289C323">3.4 <span class="Heading">Service functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X7A24C0997AC7C6A3">3.4-1 InfoCircle</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap4.html#X803553D8849F6D7A">4 <span class="Heading">A sample computation with <strong class="pkg">Circle</strong></span></a>
</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>
<br />
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