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<p>As generalizations of weak left braces proposed by Catino, Mazzotta, Miccoli, and Stefanelli (Semigroup Forum 104, 228–255 (2022)), we introduce weak left <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10539_Article_IEq1.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\star \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋆</mo> </math></EquationSource> </InlineEquation>-braces by using regular <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10539_Article_IEq1.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\star \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋆</mo> </math></EquationSource> </InlineEquation>-semigroups. We describe the algebraic structures of weak left <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10539_Article_IEq1.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\star \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋆</mo> </math></EquationSource> </InlineEquation>-braces and prove that the associated maps to weak left <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="233_2025_10539_Article_IEq1.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\star \)</EquationSource> <EquationSource Format="MATHML"><math> <mo>⋆</mo> </math></EquationSource> </InlineEquation>-braces are always set-theoretic solutions of the Yang–Baxter equation. Our work may be regarded as extending and enriching some related results on weak left braces obtained by Catino, Mazzotta, Miccoli, and Stefanelli.</p>

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Set-theoretic solutions of the Yang–Baxter equation associated to weak left \(\star \)-braces

  • 期刊论文

  • 发表日期：

    2025年5月28日

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  • Qianxue Liu,
  • Shoufeng Wang

摘要

As generalizations of weak left braces proposed by Catino, Mazzotta, Miccoli, and Stefanelli (Semigroup Forum 104, 228–255 (2022)), we introduce weak left \(\star \) ⋆ -braces by using regular \(\star \) ⋆ -semigroups. We describe the algebraic structures of weak left \(\star \) ⋆ -braces and prove that the associated maps to weak left \(\star \) ⋆ -braces are always set-theoretic solutions of the Yang–Baxter equation. Our work may be regarded as extending and enriching some related results on weak left braces obtained by Catino, Mazzotta, Miccoli, and Stefanelli.