@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:-CJ8VQBHV-S
  skos:exactMatch <https://en.wikipedia.org/wiki/Associative_property>, <https://fr.wikipedia.org/wiki/Associativit%C3%A9> ;
  skos:inScheme psr: ;
  skos:prefLabel "associative property"@en, "associativité"@fr ;
  skos:broader psr:-D02PV8F1-M ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:definition """In mathematics, the <b>associative property</b> is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, <b>associativity</b> is a valid rule of replacement for expressions in logical proofs. Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is (after rewriting the expression with parentheses and in infix notation if necessary), rearranging the parentheses in such an expression will not change its value. Consider the following equations: <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 {\\egin{aligned}(2+3)+4&amp;=2+(3+4)=9\\\\,\\\\\\\\2\\	imes (3\\	imes 4)&amp;=(2\\	imes 3)\\	imes 4=24.\\\\end{aligned}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">             <mtr>               <mtd>                 <mo stretchy="false">(</mo>                 <mn>2</mn>                 <mo>+</mo>                 <mn>3</mn>                 <mo stretchy="false">)</mo>                 <mo>+</mo>                 <mn>4</mn>               </mtd>               <mtd>                 <mi></mi>                 <mo>=</mo>                 <mn>2</mn>                 <mo>+</mo>                 <mo stretchy="false">(</mo>                 <mn>3</mn>                 <mo>+</mo>                 <mn>4</mn>                 <mo stretchy="false">)</mo>                 <mo>=</mo>                 <mn>9</mn>                 <mspace width="thinmathspace"></mspace>               </mtd>             </mtr>             <mtr>               <mtd>                 <mn>2</mn>                 <mo>×<!-- × --></mo>                 <mo stretchy="false">(</mo>                 <mn>3</mn>                 <mo>×<!-- × --></mo>                 <mn>4</mn>                 <mo stretchy="false">)</mo>               </mtd>               <mtd>                 <mi></mi>                 <mo>=</mo>                 <mo stretchy="false">(</mo>                 <mn>2</mn>                 <mo>×<!-- × --></mo>                 <mn>3</mn>                 <mo stretchy="false">)</mo>                 <mo>×<!-- × --></mo>                 <mn>4</mn>                 <mo>=</mo>                 <mn>24.</mn>               </mtd>             </mtr>           </mtable>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\egin{aligned}(2+3)+4&amp;=2+(3+4)=9\\\\,\\\\\\\\2\\	imes (3\\	imes 4)&amp;=(2\\	imes 3)\\	imes 4=24.\\\\end{aligned}}}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb094798b35a217beb6a0a79b6e72772cd585cfb" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.875ex; height:6.176ex;" alt="{\\\\displaystyle {\\egin{aligned}(2+3)+4&amp;=2+(3+4)=9\\\\,\\\\\\\\2\\	imes (3\\	imes 4)&amp;=(2\\	imes 3)\\	imes 4=24.\\\\end{aligned}}}"></div> Even though the parentheses were rearranged on each line, the values of the expressions were not altered. Since this holds true when performing addition and multiplication on any real numbers, it can be said that "addition and multiplication of real numbers are associative operations". 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Associative_property">https://en.wikipedia.org/wiki/Associative_property</a>)"""@en, """En mathématiques, et plus précisément en algèbre générale, une loi de composition interne ou loi interne <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 \\\\star }">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo>⋆<!-- ⋆ --></mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle \\\\star }</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd316a21eeb5079a850f223b1d096a06bfa788c0" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: 0.035ex; margin-bottom: -0.206ex; width:1.162ex; height:1.509ex;" alt="{\\\\displaystyle \\\\star }"></span> sur un ensemble <span class="texhtml mvar" style="font-style:italic;">E</span> est dite <b>associative</b> si pour tous <span class="texhtml mvar" style="font-style:italic;">x</span>, <span class="texhtml mvar" style="font-style:italic;">y</span> et <span class="texhtml mvar" style="font-style:italic;">z</span> dans <span class="texhtml mvar" style="font-style:italic;">E</span> :  <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 (x\\\\star y)\\\\star z=x\\\\star (y\\\\star z)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>⋆<!-- ⋆ --></mo>         <mi>y</mi>         <mo stretchy="false">)</mo>         <mo>⋆<!-- ⋆ --></mo>         <mi>z</mi>         <mo>=</mo>         <mi>x</mi>         <mo>⋆<!-- ⋆ --></mo>         <mo stretchy="false">(</mo>         <mi>y</mi>         <mo>⋆<!-- ⋆ --></mo>         <mi>z</mi>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (x\\\\star y)\\\\star z=x\\\\star (y\\\\star z)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b211723abebcb4a91912ae1fd2c3457a3980a8fa" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.643ex; height:2.843ex;" alt="{\\\\displaystyle (x\\\\star y)\\\\star z=x\\\\star (y\\\\star z)}"></span>.</dd></dl> En notant <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 m:E\\	imes E\\	o E,\\\\;(x,y)\\\\mapsto x\\\\star y}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>m</mi>         <mo>:</mo>         <mi>E</mi>         <mo>×<!-- × --></mo>         <mi>E</mi>         <mo stretchy="false">→<!-- → --></mo>         <mi>E</mi>         <mo>,</mo>         <mspace width="thickmathspace"></mspace>         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>,</mo>         <mi>y</mi>         <mo stretchy="false">)</mo>         <mo stretchy="false">↦<!-- ↦ --></mo>         <mi>x</mi>         <mo>⋆<!-- ⋆ --></mo>         <mi>y</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle m:E\\	imes E\\	o E,\\\\;(x,y)\\\\mapsto x\\\\star y}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5cb0891d3ab43faafdc74332088c13d03cafefd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.06ex; height:2.843ex;" alt="{\\\\displaystyle m:E\\	imes E\\	o E,\\\\;(x,y)\\\\mapsto x\\\\star y}"></span>, l'associativité se traduit par le diagramme commutatif suivant : Parmi les lois associatives, on peut citer les lois d'addition et de multiplication des nombres réels, des nombres complexes et des matrices carrées, l'addition des vecteurs, et l'intersection, la réunion d'ensembles. Aussi, si <var>M</var> est un ensemble quelconque et <var>S</var> désigne l'ensemble de toutes les fonctions de <var>M</var> vers <var>M</var>, alors l'opération de composition des fonctions sur <var>S</var> est associative. Parmi les lois non associatives, on peut citer par exemple le produit vectoriel sur un espace euclidien orienté de dimension 3.  Un autre exemple est la soustraction des nombres réels. En effet :   <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 (30-20)-10=10-10=0}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <mn>30</mn>         <mo>−<!-- − --></mo>         <mn>20</mn>         <mo stretchy="false">)</mo>         <mo>−<!-- − --></mo>         <mn>10</mn>         <mo>=</mo>         <mn>10</mn>         <mo>−<!-- − --></mo>         <mn>10</mn>         <mo>=</mo>         <mn>0</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (30-20)-10=10-10=0}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/950192b313ec043611b01f7903c7fa50f6aae962" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.314ex; height:2.843ex;" alt="{\\\\displaystyle (30-20)-10=10-10=0}"></span> et <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 30-(20-10)=30-10=20}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mn>30</mn>         <mo>−<!-- − --></mo>         <mo stretchy="false">(</mo>         <mn>20</mn>         <mo>−<!-- − --></mo>         <mn>10</mn>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mn>30</mn>         <mo>−<!-- − --></mo>         <mn>10</mn>         <mo>=</mo>         <mn>20</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 30-(20-10)=30-10=20}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23cf6d00b90fcca7ffc2a05d378c10a949febabd" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.477ex; height:2.843ex;" alt="{\\\\displaystyle 30-(20-10)=30-10=20}"></span></dd></dl> donc  <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 (30-20)-10\\
eq 30-(20-10)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <mn>30</mn>         <mo>−<!-- − --></mo>         <mn>20</mn>         <mo stretchy="false">)</mo>         <mo>−<!-- − --></mo>         <mn>10</mn>         <mo>≠<!-- ≠ --></mo>         <mn>30</mn>         <mo>−<!-- − --></mo>         <mo stretchy="false">(</mo>         <mn>20</mn>         <mo>−<!-- − --></mo>         <mn>10</mn>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (30-20)-10\\
eq 30-(20-10)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3179689210c03c0c6318dce0d70c9b694ac52fa4" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.028ex; height:2.843ex;" alt="{\\\\displaystyle (30-20)-10\\
eq 30-(20-10)}"></span>.</dd></dl> Un ensemble muni d'une loi interne associative et unifère est appelé un monoïde. On peut écrire un algorithme qui, pour un magma fini d'ordre <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 n}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>n</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle n}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\\\\displaystyle n}"></span> de table de Cayley donnée, détermine s'il est un groupe ou non en <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 O(n^{2})}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>O</mi>         <mo stretchy="false">(</mo>         <msup>           <mi>n</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo stretchy="false">)</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle O(n^{2})}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd9594a16cb898b8f2a2dff9227a385ec183392" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\\\\displaystyle O(n^{2})}"></span> opérations élémentaires</span>, la difficulté majeure étant de décider de l'associativité de la loi.  
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Associativit%C3%A9">https://fr.wikipedia.org/wiki/Associativit%C3%A9</a>)"""@fr ;
  a skos:Concept .

psr:-D02PV8F1-M
  skos:prefLabel "opération"@fr, "operation"@en ;
  a skos:Concept ;
  skos:narrower psr:-CJ8VQBHV-S .