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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>YangBaxter</h1>


<h2>Combinatorial Solutions for the Yang-Baxter equation</h2>

<p>
    0.10.7</p>

<p>
    14 July 2025
  </p>

</div>
<p><b>
    Leandro Vendramin




  </b>
<br />Email: <span class="URL"><a href="mailto:Leandro.Vendramin@vub.be">Leandro.Vendramin@vub.be</a></span>
<br />Homepage: <span class="URL"><a href="https://vendramin.github.io/">https://vendramin.github.io/</a></span>
<br />Address: <br />Vrije Universiteit Brussel<br /> Faculty of Sciences<br /> Department of Mathematics and Data Science<br /> Pleinlaan 2, B-1050<br /> Brussel, Belgium<br />
</p><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://olexandr-konovalov.github.io/">https://olexandr-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<br />
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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1.html#X8749E1888244CC3D">1 <span class="Heading">Preliminaries</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7BB9D67179296AA0">1.1 <span class="Heading">Definition and examples</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F66BB617A79542D">1.1-1 IsSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7C2FB9E27C641F49">1.1-2 Skewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BBF6AC978DC5CC1">1.1-3 SmallSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7B1AAF517D8D209D">1.1-4 TrivialBrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X82BB1BB37932DF70">1.1-5 TrivialSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86AFF9C586B5C2B1">1.1-6 SmallBrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7ED6436A7DC2AB48">1.1-7 IdSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7DF9D5E8817C3564">1.1-8 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X80F7E6B78327BD5E">1.1-9 IdBrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X79746AAC863EA794">1.1-10 IsomorphismSkewbraces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X87A9198C8456D193">1.1-11 DirectProductSkewbraces</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X879A94807C0A65D2">1.1-12 DirectProductOp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X84880C7484699973">1.1-13 IsTwoSided</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X84CB08C88574F438">1.1-14 IsAutomorphismGroupOfSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7BC5B7CF7877F333">1.1-15 IsClassical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7B398DBF7B2476B5">1.1-16 IsOfAbelianType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X78A3A59A86F96508">1.1-17 IsBiSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X797C4B6480DFCDDA">1.1-18 IsOfNilpotentType</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F8C6C4B81096AF0">1.1-19 IsTrivialSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8426ABE1808B92DC">1.1-20 Skewbrace2YB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7F09E6DE78CC240B">1.1-21 Brace2YB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X7E2F64338788EBF9">1.1-22 SkewbraceSubset2YB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X8439C2087DD1D9A6">1.1-23 SemidirectProduct</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X86D6131182AE2DBC">1.1-24 UnderlyingAdditiveGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1.html#X840A631685DA79D6">1.1-25 UnderlyingMultiplicativeGroup</a></span>
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<div class="ContChap"><a href="chap2.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.html#X8714568A80DBF0EF">2.1 <span class="Heading">Braces and Radical Rings</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86C2A9257D2D1CAF">2.1-1 AdditiveGroupOfRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7816FE1786837102">2.1-2 IsJacobsonRadical</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2.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.html#X7AEBEF6F7CFCA074">2.2-1 Table2YB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X825856827B8F9B3C">2.2-2 Evaluate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7EB5F8BE80E57D3E">2.2-3 LyubashenkoYB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X7B14202778611DA1">2.2-4 IsIndecomposable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X83B5B0B678E85958">2.2-5 Table</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X815F6E1287725A92">2.2-6 DehornoyClass</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X86A7FA1E843A438E">2.2-7 DehornoyRepresentationOfStructureGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X8596E3EA7E4C1067">2.2-8 IdYB</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2.html#X829BF82C814E5498">2.2-9 LinearRepresentationOfStructureGroup</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap3.html#X81D398D67DC78FB5">3 <span class="Heading">YangBaxter automatic generated documentation</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7C287494794C9DD6">3.1 <span class="Heading">YangBaxter automatic generated documentation of properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3.html#X84D5AD107FCE1467">3.1-1 IsIndecomposable</a></span>
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</div>
<div class="ContChap"><a href="chap4.html#X7FF13C7684E1122C">4 <span class="Heading">Ideals and left ideals</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X81E965A37A7EA22A">4.1 <span class="Heading">Left ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X814FEB578507E81C">4.1-1 LeftIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7AE9FAB479569BF9">4.1-2 StrongLeftIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X829DFD167A8D0D4A">4.1-3 IsLeftIdeal</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X83629803819C4A6F">4.2 <span class="Heading">Ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X879540527DA666C4">4.2-1 IsIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7EBF92377C5E417D">4.2-2 Ideals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X809F4B407D4BDE47">4.2-3 AsIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7F0A2FBA87465560">4.2-4 IdealGeneratedBy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8721D11884A2CDAD">4.2-5 IntersectionOfTwoIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85A4F7FE7B627615">4.2-6 SumOfTwoIdeals</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X8079CE3187FE380D">4.3 <span class="Heading">Sequences (left) ideals</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X845E09BF86C4DD2E">4.3-1 LeftSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B9ED49481948B91">4.3-2 RightSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X84C0A78F7B2845FD">4.3-3 IsLeftNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X79EA70287B245D65">4.3-4 IsSimpleSkewbrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D930A7679D97788">4.3-5 IsRightNilpotent</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X81593E537B94350B">4.3-6 LeftNilpotentIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7B6EB5A37EBFFB7D">4.3-7 RightNilpotentIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E9665EB79226E96">4.3-8 SmoktunowiczSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D503F497CB34B9D">4.3-9 Socle</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7FF15DAA78E08F0A">4.3-10 Annihilator</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X876342AF7CF51C9B">4.4 <span class="Heading">Mutipermutation skew braces</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7E0053787EDFEAFB">4.4-1 SocleSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85AA85F57FF7BD73">4.4-2 MultipermutationLevel</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X824956137F4CEF3C">4.4-3 IsMultipermutation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8616F73781699DC3">4.4-4 Fix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CE04CC57E82FD02">4.4-5 KernelOfLambda</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7CE55DAF7CB85B89">4.4-6 Quotient</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4.html#X7826660686D57FD6">4.5 <span class="Heading">Prime and semiprime ideals</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X82CDAD02845051FA">4.5-1 IsPrimeBrace</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X834AED5184F2B9AC">4.5-2 IsPrimeIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80D35A2880B39EB0">4.5-3 PrimeIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X820951168658A704">4.5-4 IsSemiprime</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X80961A4F7CBFBA0B">4.5-5 IsSemiprimeIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7A8C53838192CEC3">4.5-6 SemiprimeIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X7D6E642D817352AF">4.5-7 BaerRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X8571BC2F80364341">4.5-8 IsBaer</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X856E8ABD7BCA81D5">4.5-9 WedderburnRadical</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X85F4D83079E1013A">4.5-10 SolvableSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X86623B417F4F07FE">4.5-11 IsMinimalIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4.html#X837D770278330FE0">4.5-12 MinimalIdeals</a></span>
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</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>
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