@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-NGWS6NXB-P
  skos:prefLabel "octonion"@en, "octonion"@fr ;
  a skos:Concept ;
  skos:broader psr:-H2PDWGMD-8 .

psr:-ZS43QGRB-V
  skos:prefLabel "octonion déployé"@fr, "split-octonion"@en ;
  a skos:Concept ;
  skos:broader psr:-H2PDWGMD-8 .

psr:-H2PDWGMD-8
  skos:exactMatch <https://en.wikipedia.org/wiki/Alternative_algebra>, <https://fr.wikipedia.org/wiki/Alg%C3%A8bre_alternative> ;
  dc:modified "2023-08-24"^^xsd:date ;
  skos:narrower psr:-NGWS6NXB-P, psr:-HZ5JWXBQ-C, psr:-ZS43QGRB-V ;
  skos:prefLabel "alternative algebra"@en, "algèbre alternative"@fr ;
  skos:broader psr:-F1B5QL5S-0 ;
  a skos:Concept ;
  skos:definition """In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
         <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle x(xy)=(xx)y}">
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         <annotation encoding="application/x-tex">{\\\\displaystyle x(xy)=(xx)y}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205e6938b222921f0c3811037e8e615285890358" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.347ex; height:2.843ex;" alt="{\\\\displaystyle x(xy)=(xx)y}"></span></li>
         <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle (yx)x=y(xx)}">
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         <mstyle displaystyle="true" scriptlevel="0">
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         <annotation encoding="application/x-tex">{\\\\displaystyle (yx)x=y(xx)}</annotation>
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         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9868f1ec2521258187f9a99769b5f63f94ecb97" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.347ex; height:2.843ex;" alt="{\\\\displaystyle (yx)x=y(xx)}"></span></li></ul>
         for all <i>x</i> and <i>y</i> in the algebra.
<br/>Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Alternative_algebra">https://en.wikipedia.org/wiki/Alternative_algebra</a>)"""@en, """En algèbre, une algèbre alternative est une algèbre dans laquelle la multiplication n'est pas nécessairement associative mais satisfait à deux identités exprimant l'alternativité, à savoir
         <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle x(xy)=(xx)y}">
         <semantics>
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         <mstyle displaystyle="true" scriptlevel="0">
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         <mo stretchy="false">(</mo>
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         <mo>=</mo>
         <mo stretchy="false">(</mo>
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         <mi>x</mi>
         <mo stretchy="false">)</mo>
         <mi>y</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle x(xy)=(xx)y}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205e6938b222921f0c3811037e8e615285890358" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.347ex; height:2.843ex;" alt="{\\\\displaystyle x(xy)=(xx)y}"></span></li>
         <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"  alttext="{\\\\displaystyle (yx)x=y(xx)}">
         <semantics>
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         <mi>x</mi>
         <mi>x</mi>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle (yx)x=y(xx)}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9868f1ec2521258187f9a99769b5f63f94ecb97" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.347ex; height:2.843ex;" alt="{\\\\displaystyle (yx)x=y(xx)}"></span></li></ul>
         pour <i>x</i> et <i>y</i> quelconques dans l'algèbre.
<br/>Toute algèbre associative est évidemment alternative mais certaines algèbres strictement non associatives telles que les octonions le sont aussi. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_alternative">https://fr.wikipedia.org/wiki/Alg%C3%A8bre_alternative</a>)"""@fr ;
  dc:created "2023-08-24"^^xsd:date ;
  skos:inScheme psr: .

psr:-F1B5QL5S-0
  skos:prefLabel "algèbre non associative"@fr, "non-associative algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-H2PDWGMD-8 .

psr: a skos:ConceptScheme .
psr:-HZ5JWXBQ-C
  skos:prefLabel "algèbre d'octonions"@fr, "octonion algebra"@en ;
  a skos:Concept ;
  skos:broader psr:-H2PDWGMD-8 .