@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: a skos:ConceptScheme .
psr:-R556XDWR-W
  skos:prefLabel "algebraic operation"@en, "opération algébrique"@fr ;
  a skos:Concept ;
  skos:narrower psr:-FV79H6P8-S .

psr:-FV79H6P8-S
  skos:exactMatch <https://fr.wikipedia.org/wiki/Exponentiation>, <https://en.wikipedia.org/wiki/Exponentiation> ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:inScheme psr: ;
  skos:definition """En mathématiques, l’<b>exponentiation</b> est une opération binaire non commutative qui étend la notion de puissance d'un nombre en algèbre. Elle se note en plaçant l'un des opérandes en exposant (d'où son nom) de l'autre, appelé <i>base</i>. Pour des exposants rationnels, l'exponentiation est définie algébriquement de façon à satisfaire la relation :  <dl><dd><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^{b+c}=a^{b}\\	imes a^{c}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>b</mi>             <mo>+</mo>             <mi>c</mi>           </mrow>         </msup>         <mo>=</mo>         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>b</mi>           </mrow>         </msup>         <mo>×<!-- × --></mo>         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>c</mi>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a^{b+c}=a^{b}\\	imes a^{c}.}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e18c0e24c6ba09a898d24015056d5cc247e000b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.085ex; height:2.676ex;" alt="{\\\\displaystyle a^{b+c}=a^{b}\\	imes a^{c}.}"></span></dd></dl> Pour des exposants réels, complexes ou matriciels, la définition passe en général par l'utilisation de la fonction exponentielle, à condition que la base admette un logarithme :  <dl><dd><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^{b}=\\\\exp(b\\	imes \\\\ln(a)).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>a</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>b</mi>           </mrow>         </msup>         <mo>=</mo>         <mi>exp</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>b</mi>         <mo>×<!-- × --></mo>         <mi>ln</mi>         <mo>⁡<!-- ⁡ --></mo>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo stretchy="false">)</mo>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a^{b}=\\\\exp(b\\	imes \\\\ln(a)).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6bd4427d1462b90e58a9ff8aaf84f186ef8d96" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.091ex; height:3.176ex;" alt="{\\\\displaystyle a^{b}=\\\\exp(b\\	imes \\\\ln(a)).}"></span></dd></dl> L'exponentiation ensembliste est définie à l'aide des ensembles de fonctions :  <dl><dd><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 F^{E}={\\\\mathcal {F}}(E;F).}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>F</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>E</mi>           </mrow>         </msup>         <mo>=</mo>         <mrow class="MJX-TeXAtom-ORD">           <mrow class="MJX-TeXAtom-ORD">             <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi>           </mrow>         </mrow>         <mo stretchy="false">(</mo>         <mi>E</mi>         <mo>;</mo>         <mi>F</mi>         <mo stretchy="false">)</mo>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle F^{E}={\\\\mathcal {F}}(E;F).}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0626816a0cd67d5ae8075c833c7e5a9482dc724" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.334ex; height:3.176ex;" alt="{\\\\displaystyle F^{E}={\\\\mathcal {F}}(E;F).}"></span></dd></dl> Elle permet de définir l'exponentiation pour les cardinaux associés. Elle se généralise par ailleurs, en théorie des catégories, par la notion d'objet exponentiel. Enfin, l'exponentiation des ordinaux est construite par récurrence transfinie :  <dl><dd><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 \\\\alpha ^{\\eta +1}=\\\\alpha ^{\\eta }\\	imes \\\\alpha .}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>α<!-- α --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>β<!-- β --></mi>             <mo>+</mo>             <mn>1</mn>           </mrow>         </msup>         <mo>=</mo>         <msup>           <mi>α<!-- α --></mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>β<!-- β --></mi>           </mrow>         </msup>         <mo>×<!-- × --></mo>         <mi>α<!-- α --></mi>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\alpha ^{\\eta +1}=\\\\alpha ^{\\eta }\\	imes \\\\alpha .}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7461ad999592906f445877132735700988117be6" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.497ex; height:2.676ex;" alt="{\\\\displaystyle \\\\alpha ^{\\eta +1}=\\\\alpha ^{\\eta }\\	imes \\\\alpha .}"> 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Exponentiation">https://fr.wikipedia.org/wiki/Exponentiation</a>)"""@fr, """In mathematics, <b>exponentiation</b> is an operation involving two numbers: the <i>base</i> and the <i>exponent</i> or <i>power</i>. Exponentiation is written as <span class="texhtml"><i>b</i><sup><i>n</i></sup></span>, where <span class="texhtml mvar" style="font-style:italic;">b</span> is the <i>base</i> and <span class="texhtml mvar" style="font-style:italic;">n</span> is the <i>power</i>; this is pronounced as "<span class="texhtml mvar" style="font-style:italic;">b</span> (raised) to the (power of) <span class="texhtml mvar" style="font-style:italic;">n</span>". When <span class="texhtml mvar" style="font-style:italic;">n</span> is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, <span class="texhtml"><i>b</i><sup><i>n</i></sup></span> is the product of multiplying <span class="texhtml mvar" style="font-style:italic;">n</span> bases: <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle b^{n}=\\\\underbrace {b\\	imes b\\	imes \\\\dots \\	imes b\\	imes b} _{n{\\	ext{ times}}}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msup>           <mi>b</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msup>         <mo>=</mo>         <munder>           <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">             <munder>               <mrow>                 <mi>b</mi>                 <mo>×<!-- × --></mo>                 <mi>b</mi>                 <mo>×<!-- × --></mo>                 <mo>⋯<!-- ⋯ --></mo>                 <mo>×<!-- × --></mo>                 <mi>b</mi>                 <mo>×<!-- × --></mo>                 <mi>b</mi>               </mrow>               <mo>⏟<!-- ⏟ --></mo>             </munder>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mrow class="MJX-TeXAtom-ORD">               <mtext> times</mtext>             </mrow>           </mrow>         </munder>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle b^{n}=\\\\underbrace {b\\	imes b\\	imes \\\\dots \\	imes b\\	imes b} _{n{\\	ext{ times}}}.}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb8949e2c2dbe952c87d89e208b8018f107ad79" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -3.671ex; width:24.036ex; height:5.676ex;" alt="{\\\\displaystyle b^{n}=\\\\underbrace {b\\	imes b\\	imes \\\\dots \\	imes b\\	imes b} _{n{\\	ext{ times}}}.}"></div> The exponent is usually shown as a superscript to the right of the base. In that case, <span class="texhtml"><i>b</i><sup><i>n</i></sup></span> is called "<i>b</i> raised to the <i>n</i>th power", "<i>b</i> (raised) to the power of <i>n</i>", "the <i>n</i>th power of <i>b</i>", "<i>b</i> to the <i>n</i>th power", or most briefly as "<i>b</i> to the <i>n</i>(th)". 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Exponentiation">https://en.wikipedia.org/wiki/Exponentiation</a>)"""@en ;
  skos:prefLabel "exponentiation"@en, "exponentiation"@fr ;
  a skos:Concept ;
  skos:broader psr:-R556XDWR-W .