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<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap0_mj.html">[Previous Chapter]</a>    <a href="chapInd_mj.html">[Next Chapter]</a>   </div>

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<p><a id="X7EB7BB817CC15891" name="X7EB7BB817CC15891"></a></p>
<div class="ChapSects"><a href="chap1_mj.html#X7EB7BB817CC15891">1 <span class="Heading">smallantimagmas automatic generated documentation</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X78F902077A6714C4">1.1 <span class="Heading">smallantimagmas automatic generated documentation of properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B8537C57970AE37">1.1-1 IsAntiassociative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8392BFEE78284DE1">1.1-2 IsLeftCyclic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BE3D1C0801AE7EA">1.1-3 IsRightCyclic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D377A068350EE46">1.1-4 IsLeftDistributive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X856E2BBA7B87F209">1.1-5 IsRightDistributive</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D994F7382127E77">1.1-6 IsLeftCancellative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85C01ECF83CD5484">1.1-7 IsRightCancellative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E58570787350F43">1.1-8 IsCancellative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7965D94579F05D6F">1.1-9 IsLeftFPFInducted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X866768408460130D">1.1-10 IsRightFPFInducted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7B55200A7A734759">1.1-11 IsLeftDerangementInducted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E0D0CD780550E81">1.1-12 IsRightDerangementInducted</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X83147660820830E8">1.1-13 IsLeftAlternative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7C16C7657F8B5A61">1.1-14 IsRightAlternative</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X839C88FC81394B14">1.2 <span class="Heading">smallantimagmas automatic generated documentation of attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X809A477180F2A0F6">1.2-1 AssociativityIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X802DA9B083E7F4A3">1.2-2 DiagonalOfMultiplicationTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87FED7267EB0A068">1.2-3 CommutativityIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D32B8F780294AEC">1.2-4 AnticommutativityIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X85904967802C27B4">1.2-5 SquaresIndex</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X783C01BD7B931A12">1.2-6 IdSmallAntimagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X78BF96667D805695">1.2-7 LeftOrder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X850894387AD0E717">1.2-8 RightOrder</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7CDCE61679BFD1F7">1.2-9 LeftOrdersOfElements</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X84D8A5D88790F870">1.2-10 RightOrdersOfElements</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X7A4C95527CCD8524">1.3 <span class="Heading">smallantimagmas automatic generated documentation of global functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X8011AFA47C662D53">1.3-1 AllSubmagmas</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7BA7BE44861A8C51">1.3-2 MagmaIsomorphismInvariantsMatch</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7F7EAF23807EE984">1.3-3 IsMagmaIsomorphic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X807BD59180325341">1.3-4 IsMagmaAntiisomorphic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X79BF9EB48562EF6C">1.3-5 TransposedMagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7FD21A58851687DF">1.3-6 LeftPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X788005DE7947D914">1.3-7 RightPower</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7DDB92C284197B0B">1.3-8 AllSmallAntimagmas</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7E5054007A985873">1.3-9 NrSmallAntimagmas</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7903CEEA79DEBA73">1.3-10 SmallAntimagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X804A623879A9AC7F">1.3-11 OneSmallAntimagma</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7D0A720279246D45">1.3-12 ReallyAllSmallAntimagmas</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X796637787C216BA1">1.3-13 ReallyNrSmallAntimagmas</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X8553DA53796C81EA">1.4 <span class="Heading">smallantimagmas automatic generated documentation of methods</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X87F3EA827FF3D302">1.4-1 MagmaIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X7AB3F3E37D8A5687">1.4-2 MagmaAntiisomorphism</a></span>
</div></div>
</div>

<h3>1 <span class="Heading">smallantimagmas automatic generated documentation</span></h3>

<p><a id="X78F902077A6714C4" name="X78F902077A6714C4"></a></p>

<h4>1.1 <span class="Heading">smallantimagmas automatic generated documentation of properties</span></h4>

<p><a id="X7B8537C57970AE37" name="X7B8537C57970AE37"></a></p>

<h5>1.1-1 IsAntiassociative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAntiassociative</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>identifies whether magma <var class="Arg">M</var> is antiassociative.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAntiassociative(OneSmallGroup(16));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAntiassociative(OneSmallAntimagma(2));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAntiassociative(OneSmallAntimagma(3));</span>
true
</pre></div>

<p><a id="X8392BFEE78284DE1" name="X8392BFEE78284DE1"></a></p>

<h5>1.1-2 IsLeftCyclic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftCyclic</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is left cyclic <var class="Arg">m</var>.</p>

<p><a id="X7BE3D1C0801AE7EA" name="X7BE3D1C0801AE7EA"></a></p>

<h5>1.1-3 IsRightCyclic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightCyclic</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is left cyclic <var class="Arg">m</var>.</p>

<p><a id="X7D377A068350EE46" name="X7D377A068350EE46"></a></p>

<h5>1.1-4 IsLeftDistributive</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftDistributive</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is left distributive <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(3), M -> IsLeftDistributive(M) );</span>
[ true, false, false, false, false, false, false, false, false, true ]
</pre></div>

<p><a id="X856E2BBA7B87F209" name="X856E2BBA7B87F209"></a></p>

<h5>1.1-5 IsRightDistributive</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightDistributive</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is right distributive <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(3), M -> IsRightDistributive(M) );</span>
[ false, false, false, false, true, false, false, false, true, false ]
</pre></div>

<p><a id="X7D994F7382127E77" name="X7D994F7382127E77"></a></p>

<h5>1.1-6 IsLeftCancellative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftCancellative</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is left cancellative <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( MultiplicationTable(M) );</span>
[ [  2,  1 ],
  [  2,  1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightCancellative(M);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftCancellative(M);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(2), M -> IsLeftCancellative(M));</span>
[ true, false ]
</pre></div>

<p><a id="X85C01ECF83CD5484" name="X85C01ECF83CD5484"></a></p>

<h5>1.1-7 IsRightCancellative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightCancellative</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is right cancellative <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(2), M -> IsRightCancellative(M));</span>
[ false, true ]
</pre></div>

<p><a id="X7E58570787350F43" name="X7E58570787350F43"></a></p>

<h5>1.1-8 IsCancellative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsCancellative</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>if magma is cancellative <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(2), M -> IsCancellative(M));</span>
[ false, false ]
</pre></div>

<p><a id="X7965D94579F05D6F" name="X7965D94579F05D6F"></a></p>

<h5>1.1-9 IsLeftFPFInducted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftFPFInducted</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>is a left-hand sided fixed-point free inducted <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( MultiplicationTable( SmallAntimagma(2, 2) ) );</span>
[ [  2,  2 ],
  [  1,  1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftFPFInducted( SmallAntimagma(2, 2) ); </span>
true
</pre></div>

<p><a id="X866768408460130D" name="X866768408460130D"></a></p>

<h5>1.1-10 IsRightFPFInducted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightFPFInducted</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>is a right-hand sided fixed-point free inducted <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( MultiplicationTable( SmallAntimagma(2, 1) ) );</span>
[ [  2,  1 ],
  [  2,  1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightFPFInducted( SmallAntimagma(2, 1) );            </span>
true
</pre></div>

<p><a id="X7B55200A7A734759" name="X7B55200A7A734759"></a></p>

<h5>1.1-11 IsLeftDerangementInducted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftDerangementInducted</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>is a left-hand sided derangment inducted <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 2);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftFPFInducted(M);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightFPFInducted(M);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightDerangementInducted(M);</span>
false
</pre></div>

<p><a id="X7E0D0CD780550E81" name="X7E0D0CD780550E81"></a></p>

<h5>1.1-12 IsRightDerangementInducted</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightDerangementInducted</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>is a right-hand sided derangment inducted <var class="Arg">m</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsLeftFPFInducted(M);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightFPFInducted(M);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsRightDerangementInducted(M);</span>
true
</pre></div>

<p><a id="X83147660820830E8" name="X83147660820830E8"></a></p>

<h5>1.1-13 IsLeftAlternative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLeftAlternative</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>is a left-alternatve magma <var class="Arg">M</var>.</p>


<div class="example"><pre>
</pre></div>

<p><a id="X7C16C7657F8B5A61" name="X7C16C7657F8B5A61"></a></p>

<h5>1.1-14 IsRightAlternative</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRightAlternative</code>( <var class="Arg">M</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: <code class="keyw">true</code> or <code class="keyw">false</code></p>

<p>is a right-alternatve magma <var class="Arg">M</var>.</p>


<div class="example"><pre>
</pre></div>

<p><a id="X839C88FC81394B14" name="X839C88FC81394B14"></a></p>

<h4>1.2 <span class="Heading">smallantimagmas automatic generated documentation of attributes</span></h4>

<p><a id="X809A477180F2A0F6" name="X809A477180F2A0F6"></a></p>

<h5>1.2-1 AssociativityIndex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AssociativityIndex</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>identifies associativity index of <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSmallAntimagma(2);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">AssociativityIndex(OneSmallAntimagma(2));</span>
0
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSmallGroup(4);</span>
<pc group of size 4 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">AssociativityIndex(OneSmallGroup(4));</span>
64
<span class="GAPprompt">gap></span> <span class="GAPinput">AssociativityIndex(OneSmallGroup(4)) = 4 ^ 3;</span>
true
</pre></div>

<p><a id="X802DA9B083E7F4A3" name="X802DA9B083E7F4A3"></a></p>

<h5>1.2-2 DiagonalOfMultiplicationTable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DiagonalOfMultiplicationTable</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes diaognal of multiplication table of <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(3), M -> DiagonalOfMultiplicationTable((M)));                </span>
[ [ 2, 1, 1 ], [ 2, 1, 1 ], 
  [ 2, 3, 2 ], [ 2, 1, 1 ], 
  [ 2, 1, 1 ], [ 2, 1, 2 ], 
  [ 2, 3, 2 ], [ 2, 1, 2 ], 
  [ 2, 3, 1 ], [ 2, 3, 1 ]
]
</pre></div>

<p><a id="X87FED7267EB0A068" name="X87FED7267EB0A068"></a></p>

<h5>1.2-3 CommutativityIndex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CommutativityIndex</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>identifies commutativity index of <var class="Arg">M</var>.</p>


<div class="example"><pre>

</pre></div>

<p><a id="X7D32B8F780294AEC" name="X7D32B8F780294AEC"></a></p>

<h5>1.2-4 AnticommutativityIndex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AnticommutativityIndex</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>calculates anticommutativity index of <var class="Arg">M</var>.</p>


<div class="example"><pre>
</pre></div>

<p><a id="X85904967802C27B4" name="X85904967802C27B4"></a></p>

<h5>1.2-5 SquaresIndex</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SquaresIndex</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>computes squares index of <var class="Arg">M</var> so the order of <span class="SimpleMath">\(\left\{ m^2 | m \in M \right\}\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(2), M -> List(M, m -> m * m) );                </span>
[ [ m2, m1 ], [ m2, m1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(2), M -> SquaresIndex(M ));</span>
[ 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(3), M -> SquaresIndex(M ));</span>
[ 2, 2, 2, 2, 2, 2, 2, 2, 3, 3 ]
</pre></div>

<p><a id="X783C01BD7B931A12" name="X783C01BD7B931A12"></a></p>

<h5>1.2-6 IdSmallAntimagma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdSmallAntimagma</code>( <var class="Arg">M</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>identifies class of antiassociative magma <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAntiassociative(OneSmallGroup(16));</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAntiassociative(OneSmallAntimagma(2));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAntiassociative(OneSmallAntimagma(3));</span>
true
</pre></div>

<p><a id="X78BF96667D805695" name="X78BF96667D805695"></a></p>

<h5>1.2-7 LeftOrder</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftOrder</code>( [<var class="Arg">m</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a left order of element <var class="Arg">m</var>.</p>

<p><a id="X850894387AD0E717" name="X850894387AD0E717"></a></p>

<h5>1.2-8 RightOrder</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightOrder</code>( [<var class="Arg">m</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a right order of element <var class="Arg">m</var>.</p>

<p><a id="X7CDCE61679BFD1F7" name="X7CDCE61679BFD1F7"></a></p>

<h5>1.2-9 LeftOrdersOfElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftOrdersOfElements</code>( [<var class="Arg">m</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a left order of element <var class="Arg">m</var>.</p>

<p><a id="X84D8A5D88790F870" name="X84D8A5D88790F870"></a></p>

<h5>1.2-10 RightOrdersOfElements</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightOrdersOfElements</code>( [<var class="Arg">m</var>] )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns a left order of element <var class="Arg">m</var>.</p>

<p><a id="X7A4C95527CCD8524" name="X7A4C95527CCD8524"></a></p>

<h4>1.3 <span class="Heading">smallantimagmas automatic generated documentation of global functions</span></h4>

<p><a id="X8011AFA47C662D53" name="X8011AFA47C662D53"></a></p>

<h5>1.3-1 AllSubmagmas</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSubmagmas</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>builds a collection of non-isomorphic submagmas of <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallAntimagmas(2);</span>
[ <magma with 2 generators>, <magma with 2 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">List(AllSmallAntimagmas(2), M -> AllSubmagmas(M));</span>
[ [ <magma with 1 generator> ], [ <magma with 1 generator> ] ]
</pre></div>

<p><a id="X7BA7BE44861A8C51" name="X7BA7BE44861A8C51"></a></p>

<h5>1.3-2 MagmaIsomorphismInvariantsMatch</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MagmaIsomorphismInvariantsMatch</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>computes isomorphism invariants of <var class="Arg">M</var>.</p>

<p><a id="X7F7EAF23807EE984" name="X7F7EAF23807EE984"></a></p>

<h5>1.3-3 IsMagmaIsomorphic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMagmaIsomorphic</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>identifies whether magmas <var class="Arg">M</var>, <var class="Arg">N</var> are isomorphic.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := SmallAntimagma(2, 2);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">T := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaIsomorphic(M, M);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaIsomorphic(M, T);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaIsomorphic(M, N);</span>
false
</pre></div>

<p><a id="X807BD59180325341" name="X807BD59180325341"></a></p>

<h5>1.3-4 IsMagmaAntiisomorphic</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsMagmaAntiisomorphic</code>( [<var class="Arg">M</var>, <var class="Arg">N</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>identifies whether magmas <var class="Arg">M</var>, <var class="Arg">N</var> are antiisomorphic.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := SmallAntimagma(2, 2);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaAntiisomorphic(M, M);</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaAntiisomorphic(M, N);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaAntiisomorphic(M, TransposedMagma(M));</span>
true
</pre></div>

<p><a id="X79BF9EB48562EF6C" name="X79BF9EB48562EF6C"></a></p>

<h5>1.3-5 TransposedMagma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TransposedMagma</code>( [<var class="Arg">M</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>generates transposed magma <var class="Arg">M</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaAntiisomorphic(M, TransposedMagma(M));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsMagmaIsomorphic(M, TransposedMagma(TransposedMagma(M)));</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(MultiplicationTable(M));</span>
[ [  2,  1 ],
  [  2,  1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(MultiplicationTable(TransposedMagma(M)));</span>
[ [  2,  2 ],
  [  1,  1 ] ]
</pre></div>

<p><a id="X7FD21A58851687DF" name="X7FD21A58851687DF"></a></p>

<h5>1.3-6 LeftPower</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeftPower</code>( [<var class="Arg">m</var>, <var class="Arg">k</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a left <span class="SimpleMath">\(k\)</span>-power of element <var class="Arg">m</var>.</p>

<p><a id="X788005DE7947D914" name="X788005DE7947D914"></a></p>

<h5>1.3-7 RightPower</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RightPower</code>( [<var class="Arg">m</var>, <var class="Arg">k</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a right <span class="SimpleMath">\(k\)</span>-power of element <var class="Arg">m</var>.</p>

<p><a id="X7DDB92C284197B0B" name="X7DDB92C284197B0B"></a></p>

<h5>1.3-8 AllSmallAntimagmas</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSmallAntimagmas</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns all antiassociative magmas of specified size <var class="Arg">n</var> (a number)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallAntimagmas(2);</span>
[ <magma with 2 generators>, <magma with 2 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallAntimagmas(3);</span>
[ 
  <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>,
  <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>,
  <magma with 3 generators>, <magma with 3 generators>,
  <magma with 3 generators>, <magma with 3 generators>
]
</pre></div>

<p><a id="X7E5054007A985873" name="X7E5054007A985873"></a></p>

<h5>1.3-9 NrSmallAntimagmas</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NrSmallAntimagmas</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>counts number of antiassociative magmas of specified size <var class="Arg">n</var> (a number).</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NrSmallAntimagmas(2);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">NrSmallAntimagmas(3);</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">NrSmallAntimagmas(4);</span>
17780
</pre></div>

<p><a id="X7903CEEA79DEBA73" name="X7903CEEA79DEBA73"></a></p>

<h5>1.3-10 SmallAntimagma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallAntimagma</code>( <var class="Arg">n</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns antiassociative magma of id <var class="Arg">[n, i]</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallAntimagma(4, 5);</span>
<magma with 4 generators>
</pre></div>

<p><a id="X804A623879A9AC7F" name="X804A623879A9AC7F"></a></p>

<h5>1.3-11 OneSmallAntimagma</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSmallAntimagma</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns a random antiassociative magma of size <var class="Arg">n</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">OneSmallAntimagma(2);</span>
<magma with 2 generators>

<span class="GAPprompt">gap></span> <span class="GAPinput">OneSmallAntimagma(3);</span>
<magma with 3 generators>
</pre></div>

<p><a id="X7D0A720279246D45" name="X7D0A720279246D45"></a></p>

<h5>1.3-12 ReallyAllSmallAntimagmas</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReallyAllSmallAntimagmas</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns really-all antiassociative magmas, isomorphic, of specified size <var class="Arg">n</var> (a number)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReallyAllSmallAntimagmas(2);</span>
[ <magma with 2 generators>, <magma with 2 generators> ]
</pre></div>

<p><a id="X796637787C216BA1" name="X796637787C216BA1"></a></p>

<h5>1.3-13 ReallyNrSmallAntimagmas</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReallyNrSmallAntimagmas</code>( <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>counts number of antiassociative magmas of specified size <var class="Arg">n</var> (a number)</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ReallyNrSmallAntimagmas(3);</span>
52
</pre></div>

<p><a id="X8553DA53796C81EA" name="X8553DA53796C81EA"></a></p>

<h4>1.4 <span class="Heading">smallantimagmas automatic generated documentation of methods</span></h4>

<p><a id="X87F3EA827FF3D302" name="X87F3EA827FF3D302"></a></p>

<h5>1.4-1 MagmaIsomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MagmaIsomorphism</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>computes an isomoprhism between magmas <var class="Arg">M</var>, <var class="Arg">N</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := MagmaByMultiplicationTable([ [2, 1], [2, 1] ]);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MagmaIsomorphism(M, N);</span>
<general mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
</pre></div>

<p><a id="X7AB3F3E37D8A5687" name="X7AB3F3E37D8A5687"></a></p>

<h5>1.4-2 MagmaAntiisomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MagmaAntiisomorphism</code>( <var class="Arg">M</var>, <var class="Arg">N</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>creates an antiisomoprhism between magmas <var class="Arg">M</var>, <var class="Arg">N</var>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">M := SmallAntimagma(2, 1);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">N := SmallAntimagma(2, 2);</span>
<magma with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">MagmaAntiisomorphism(M, N);</span>
<mapping: Domain([ m1, m2 ]) -> Domain([ m1, m2 ]) >
</pre></div>


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