@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .

psr:-LLND57KL-D
  skos:prefLabel "algèbre associative"@fr, "associative algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-Q3PWR5P9-R .

psr:-BW0R0X6G-X
  skos:prefLabel "Grassmann number"@en, "nombre de Grassmann"@fr ;
  a skos:Concept ;
  skos:broader psr:-Q3PWR5P9-R .

psr:-Q3PWR5P9-R
  skos:broader psr:-LLND57KL-D ;
  skos:altLabel "algèbre de Grassmann"@fr, "Grassmann algebra"@en ;
  skos:prefLabel "algèbre extérieure"@fr, "exterior algebra"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Exterior_algebra>, <https://fr.wikipedia.org/wiki/Alg%C3%A8bre_ext%C3%A9rieure> ;
  skos:definition """En mathématiques, et plus précisément en algèbre et en analyse vectorielle, l'algèbre extérieure d'un espace vectoriel <i>E</i> est une algèbre associative graduée, notée <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 \\\\Lambda E}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi mathvariant="normal">Λ<!-- Λ --></mi>
         <mi>E</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Lambda E}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5542664e3db0ccb871dff76ae775e4aea947db2" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.389ex; height:2.176ex;" alt="\\\\Lambda E"></span>. La multiplication entre deux éléments <i>a</i> et <i>b</i> est appelée le <b>produit extérieur</b> et est notée  <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 a\\\\wedge b}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>∧<!-- ∧ --></mo>
         <mi>b</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\wedge b}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1dc496a5b5da3e9b94eb72f04a54167dfe022e45" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.81ex; height:2.176ex;" alt="{\\\\displaystyle a\\\\wedge b}"></span>. Le carré de tout élément de E est zéro (<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 a\\\\wedge a=0}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>∧<!-- ∧ --></mo>
         <mi>a</mi>
         <mo>=</mo>
         <mn>0</mn>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\wedge a=0}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d08eba41ec475a3174f65c223f567111c822be0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.303ex; height:2.176ex;" alt="{\\\\displaystyle a\\\\wedge a=0}"></span>), on dit que la multiplication est alternée, ce qui entraîne que pour deux éléments de E : <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 a\\\\wedge b=-b\\\\wedge a}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>∧<!-- ∧ --></mo>
         <mi>b</mi>
         <mo>=</mo>
         <mo>−<!-- − --></mo>
         <mi>b</mi>
         <mo>∧<!-- ∧ --></mo>
         <mi>a</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\wedge b=-b\\\\wedge a}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae9dad340def0f6bd8648f4a3560a2b29fb7d9fd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.527ex; height:2.343ex;" alt="{\\\\displaystyle a\\\\wedge b=-b\\\\wedge a}"></span> (la loi est «anti-commutative»).
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre_ext%C3%A9rieure">https://fr.wikipedia.org/wiki/Alg%C3%A8bre_ext%C3%A9rieure</a>)"""@fr, """In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and  v, denoted by u ∧ v, is called a bivector and lives in a space called the exterior square, a vector space that is distinct from the original space of vectors. The magnitude of u ∧ v can be interpreted as the area of the parallelogram with sides u and  v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning that u ∧ v = − ( v ∧ u ) for all vectors u and  v, but, unlike the cross product, the exterior product is associative (after introducing the exterior cubic, that is, oriented volume). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Exterior_algebra">https://en.wikipedia.org/wiki/Exterior_algebra</a>)"""@en ;
  skos:narrower psr:-BW0R0X6G-X, psr:-HZ85PFTH-T ;
  a skos:Concept ;
  skos:inScheme psr: .

psr: a skos:ConceptScheme .
psr:-HZ85PFTH-T
  skos:prefLabel "exterior product"@en, "produit extérieur"@fr ;
  a skos:Concept ;
  skos:broader psr:-Q3PWR5P9-R .