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<p><a id="X8237B3628443C3FA" name="X8237B3628443C3FA"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X8237B3628443C3FA">2 <span class="Heading">Algebraic Properties of Braces</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X8714568A80DBF0EF">2.1 <span class="Heading">Braces and Radical Rings</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X86C2A9257D2D1CAF">2.1-1 AdditiveGroupOfRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7816FE1786837102">2.1-2 IsJacobsonRadical</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X80AF1831874915EB">2.2 <span class="Heading">Braces and Yang-Baxter Equation</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7AEBEF6F7CFCA074">2.2-1 Table2YB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X825856827B8F9B3C">2.2-2 Evaluate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7EB5F8BE80E57D3E">2.2-3 LyubashenkoYB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7B14202778611DA1">2.2-4 IsIndecomposable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X83B5B0B678E85958">2.2-5 Table</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X815F6E1287725A92">2.2-6 DehornoyClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X86A7FA1E843A438E">2.2-7 DehornoyRepresentationOfStructureGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X8596E3EA7E4C1067">2.2-8 IdYB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X829BF82C814E5498">2.2-9 LinearRepresentationOfStructureGroup</a></span>
</div></div>
</div>

<h3>2 <span class="Heading">Algebraic Properties of Braces</span></h3>

<p><a id="X8714568A80DBF0EF" name="X8714568A80DBF0EF"></a></p>

<h4>2.1 <span class="Heading">Braces and Radical Rings</span></h4>

<p><a id="X86C2A9257D2D1CAF" name="X86C2A9257D2D1CAF"></a></p>

<h5>2.1-1 AdditiveGroupOfRing</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AdditiveGroupOfRing</code>( <var class="Arg">ring</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a group</p>

<p>This function returns a permutation representation of the additive group of the given ring.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">rg := SmallRing(8,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(AdditiveGroupOfRing(rg));</span>
"C4 x C2"
</pre></div>

<p><a id="X7816FE1786837102" name="X7816FE1786837102"></a></p>

<h5>2.1-2 IsJacobsonRadical</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsJacobsonRadical</code>( <var class="Arg">ring</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: true if the ring is radical and false otherwise.</p>

<p>This function checks whether a ring is Jacobson radical.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">rg := SmallRing(8,11);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsJacobsonRadical(rg);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">rg := SmallRing(8,20);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsJacobsonRadical(rg);</span>
false
</pre></div>

<p><a id="X80AF1831874915EB" name="X80AF1831874915EB"></a></p>

<h4>2.2 <span class="Heading">Braces and Yang-Baxter Equation</span></h4>

<p><a id="X7AEBEF6F7CFCA074" name="X7AEBEF6F7CFCA074"></a></p>

<h5>2.2-1 Table2YB</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Table2YB</code>( <var class="Arg">table</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: the solution given by the table</p>

<p>Given the table with <span class="SimpleMath">\(r(x,y)\)</span> in the position <span class="SimpleMath">\((x,y)\)</span> find the corresponding <span class="SimpleMath">\(r\)</span></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">l := Table(SmallIYB(4,13));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">t := Table2YB(l);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdCycleSet(YB2CycleSet(t));</span>
[ 4, 13 ]
</pre></div>

<p><a id="X825856827B8F9B3C" name="X825856827B8F9B3C"></a></p>

<h5>2.2-2 Evaluate</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Evaluate</code>( <var class="Arg">obj</var>, <var class="Arg">pair</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a pair of two integers</p>

<p>Given the pair <span class="SimpleMath">\((x,y)\)</span> this function returns <span class="SimpleMath">\(r(x,y)\)</span>.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">cs := SmallCycleSet(4,13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := CycleSet2YB(cs);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Permutations(yb);</span>
[ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ], 
  [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Evaluate(yb, [1,2]);</span>
[ 2, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Evaluate(yb, [1,3]); </span>
[ 4, 2 ]
</pre></div>

<p><a id="X7EB5F8BE80E57D3E" name="X7EB5F8BE80E57D3E"></a></p>

<h5>2.2-3 LyubashenkoYB</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LyubashenkoYB</code>( <var class="Arg">size</var>, <var class="Arg">f</var>, <var class="Arg">g</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a permutation solution to the YBE</p>

<p>Finite Lyubashenko (or permutation) solutions are defined as follows: Let <span class="SimpleMath">\(X=\{1,\dots,n\}\)</span> and <span class="SimpleMath">\(f,g\colon X\to X\)</span> be bijective functions such that <span class="SimpleMath">\(fg=gf\)</span>. Then <span class="SimpleMath">\((X,r)\)</span>, where <span class="SimpleMath">\(r(x,y)=(f(y),g(x))\)</span>, is a set-theoretic solution to the YBE.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := LyubashenkoYB(4, (1,2),(3,4));</span>
<A set-theoretical solution of size 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">Permutations(last);</span>
[ [ (1,2), (1,2), (1,2), (1,2) ], [ (3,4), (3,4), (3,4), (3,4) ] ]
</pre></div>

<p><a id="X7B14202778611DA1" name="X7B14202778611DA1"></a></p>

<h5>2.2-4 IsIndecomposable</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsIndecomposable</code>( <var class="Arg">X</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns: true if the involutive solutions is indecomposable</p>

<p><a id="X83B5B0B678E85958" name="X83B5B0B678E85958"></a></p>

<h5>2.2-5 Table</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Table</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: a table with the image of the solution</p>

<p>The table shows the value of <span class="SimpleMath">\(r(x,y)\)</span> for each <span class="SimpleMath">\((x,y)\)</span></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := SmallIYB(3,2);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Table(yb);</span>
[ [ [ 1, 1 ], [ 2, 1 ], [ 3, 2 ] ], [ [ 1, 2 ], [ 2, 2 ], [ 3, 1 ] ], [ [ 2, 3 ], [ 1, 3 ], [ 3, 3 ] ] ]
</pre></div>

<p><a id="X815F6E1287725A92" name="X815F6E1287725A92"></a></p>

<h5>2.2-6 DehornoyClass</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DehornoyClass</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: The class of an involutive solution</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">cs := SmallCycleSet(4,13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := CycleSet2YB(cs);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DehornoyClass(yb);</span>
2
<span class="GAPprompt">gap></span> <span class="GAPinput">cs := SmallCycleSet(4,19);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := CycleSet2YB(cs);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DehornoyClass(yb);</span>
4
</pre></div>

<p><a id="X86A7FA1E843A438E" name="X86A7FA1E843A438E"></a></p>

<h5>2.2-7 DehornoyRepresentationOfStructureGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DehornoyRepresentationOfStructureGroup</code>( <var class="Arg">obj</var>, <var class="Arg">variable</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: A faithful linear representation of the structure group of <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">cs := SmallCycleSet(4,13);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := CycleSet2YB(cs);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Permutations(yb);</span>
[ [ (3,4), (1,3,2,4), (1,4,2,3), (1,2) ], 
  [ (2,4), (1,4,3,2), (1,2,3,4), (1,3) ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">field := FunctionField(Rationals, 1);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">q := IndeterminatesOfFunctionField(field)[1];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := DehornoyRepresentationOfStructureGroup(yb, q);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x1 := G.1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x2 := G.2;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x3 := G.3;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x4 := G.4;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">x1*x2=x2*x4;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x1*x3=x4*x2;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x1*x4=x3*x3;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x2*x1=x3*x4;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x2*x2=x4*x1;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">x3*x1=x4*x3;</span>
true
</pre></div>

<p><a id="X8596E3EA7E4C1067" name="X8596E3EA7E4C1067"></a></p>

<h5>2.2-8 IdYB</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdYB</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the identification number of <var class="Arg">obj</var></p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">cs := SmallCycleSet(5,10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdCycleSet(cs);</span>
[ 5, 10 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">cs := SmallCycleSet(4,3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := CycleSet2YB(cs);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdYB(yb);</span>
[ 4, 3 ]
</pre></div>

<p><a id="X829BF82C814E5498" name="X829BF82C814E5498"></a></p>

<h5>2.2-9 LinearRepresentationOfStructureGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LinearRepresentationOfStructureGroup</code>( <var class="Arg">obj</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the permutation brace of the involutive solution of <var class="Arg">obj</var> a linear representation of the structure group of a finite involutive solution</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := SmallIYB(5,86);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdBrace(IYBBrace(yb));</span>
[ 6, 2 ]
</pre></div>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">yb := SmallIYB(5,86);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gr := LinearRepresentationOfStructureGroup(yb);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens := GeneratorsOfGroup(gr);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display(gens[1]);</span>
[ [  0,  1,  0,  0,  0,  1 ],
  [  1,  0,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  1,  0 ],
  [  0,  0,  1,  0,  0,  0 ],
  [  0,  0,  0,  1,  0,  0 ],
  [  0,  0,  0,  0,  0,  1 ] ]
</pre></div>


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