@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:-RS9D4FLP-X
  skos:broader psr:-SKGJ9CKK-N, psr:-LMDZ11CG-L, psr:-CTGXK4K4-T ;
  a skos:Concept ;
  dc:modified "2023-08-23"^^xsd:date ;
  skos:prefLabel "homotopy associative algebra"@en, "algèbre associative d'homotopie"@fr ;
  dc:created "2023-08-23"^^xsd:date ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Homotopy_associative_algebra> ;
  skos:inScheme psr: ;
  skos:definition """In mathematics, an algebra such as <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 (\\\\mathbb {R} ,+,\\\\cdot )}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo stretchy="false">(</mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         <mo>,</mo>
         <mo>+</mo>
         <mo>,</mo>
         <mo>⋅<!-- ⋅ --></mo>
         <mo stretchy="false">)</mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle (\\\\mathbb {R} ,+,\\\\cdot )}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed3304e521dac824c9b33476be16be5fadde282" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.01ex; height:2.843ex;" alt="{\\\\displaystyle (\\\\mathbb {R} ,+,\\\\cdot )}"></span> has multiplication <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 \\\\cdot }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mo>⋅<!-- ⋅ --></mo>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle \\\\cdot }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba2c023bad1bd39ed49080f729cbf26bc448c9ba" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: 0.439ex; margin-bottom: -0.61ex; width:0.647ex; height:1.176ex;" alt="\\\\cdot "></span> whose associativity is well-defined on the nose. This means for any real numbers <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\\\\in \\\\mathbb {R} }">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo>,</mo>
         <mi>c</mi>
         <mo>∈<!-- ∈ --></mo>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="double-struck">R</mi>
         </mrow>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a,b,c\\\\in \\\\mathbb {R} }</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5db5a5687f2ba931e229a9a3c402f70bb49aad61" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.821ex; height:2.509ex;" alt="{\\\\displaystyle a,b,c\\\\in \\\\mathbb {R} }"></span> we have
         
         <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\\\\cdot (b\\\\cdot c)-(a\\\\cdot b)\\\\cdot c=0}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>⋅<!-- ⋅ --></mo>
         <mo stretchy="false">(</mo>
         <mi>b</mi>
         <mo>⋅<!-- ⋅ --></mo>
         <mi>c</mi>
         <mo stretchy="false">)</mo>
         <mo>−<!-- − --></mo>
         <mo stretchy="false">(</mo>
         <mi>a</mi>
         <mo>⋅<!-- ⋅ --></mo>
         <mi>b</mi>
         <mo stretchy="false">)</mo>
         <mo>⋅<!-- ⋅ --></mo>
         <mi>c</mi>
         <mo>=</mo>
         <mn>0</mn>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\cdot (b\\\\cdot c)-(a\\\\cdot b)\\\\cdot c=0}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff059bd74978f1ea169ce9cc3df92752d3f2f3e1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.905ex; height:2.843ex;" alt="{\\\\displaystyle a\\\\cdot (b\\\\cdot c)-(a\\\\cdot b)\\\\cdot c=0}"></span>.</dd></dl>
         But, there are algebras <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 R}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>R</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle R}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="R"></span> which are not necessarily associative, meaning if <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\\\\in R}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>,</mo>
         <mi>b</mi>
         <mo>,</mo>
         <mi>c</mi>
         <mo>∈<!-- ∈ --></mo>
         <mi>R</mi>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a,b,c\\\\in R}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/155a175e62376943eefca56b9596ea133f93fe1c" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.907ex; height:2.509ex;" alt="{\\\\displaystyle a,b,c\\\\in R}"></span> then
         
         <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\\\\cdot (b\\\\cdot c)-(a\\\\cdot b)\\\\cdot c\\
eq 0}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <mi>a</mi>
         <mo>⋅<!-- ⋅ --></mo>
         <mo stretchy="false">(</mo>
         <mi>b</mi>
         <mo>⋅<!-- ⋅ --></mo>
         <mi>c</mi>
         <mo stretchy="false">)</mo>
         <mo>−<!-- − --></mo>
         <mo stretchy="false">(</mo>
         <mi>a</mi>
         <mo>⋅<!-- ⋅ --></mo>
         <mi>b</mi>
         <mo stretchy="false">)</mo>
         <mo>⋅<!-- ⋅ --></mo>
         <mi>c</mi>
         <mo>≠<!-- ≠ --></mo>
         <mn>0</mn>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\cdot (b\\\\cdot c)-(a\\\\cdot b)\\\\cdot c\\
eq 0}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10330470585f3d402cfd27df52a1cf8b6823ca33" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.905ex; height:2.843ex;" alt="{\\\\displaystyle a\\\\cdot (b\\\\cdot c)-(a\\\\cdot b)\\\\cdot c\\
eq 0}"></span></dd></dl>
         in general. There is a notion of algebras, called <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_{\\\\infty }}">
         <semantics>
         <mrow class="MJX-TeXAtom-ORD">
         <mstyle displaystyle="true" scriptlevel="0">
         <msub>
         <mi>A</mi>
         <mrow class="MJX-TeXAtom-ORD">
         <mi mathvariant="normal">∞<!-- ∞ --></mi>
         </mrow>
         </msub>
         </mstyle>
         </mrow>
         <annotation encoding="application/x-tex">{\\\\displaystyle A_{\\\\infty }}</annotation>
         </semantics>
         </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15b6dfe5968776343496f22a0a90c8406065def1" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.619ex; height:2.509ex;" alt="A_{\\\\infty }"></span>-algebras, which still have a property on the multiplication which still acts like the first relation, meaning associativity holds, but only holds up to a homotopy, which is a way to say after an operation "compressing" the information in the algebra, the multiplication is associative. This means although we get something which looks like the second equation, the one of inequality, we actually get equality after "compressing" the information in the algebra.
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Homotopy_associative_algebra">https://en.wikipedia.org/wiki/Homotopy_associative_algebra</a>)"""@en .

psr:-CTGXK4K4-T
  skos:prefLabel "théorie de l'homotopie"@fr, "homotopy theory"@en ;
  a skos:Concept ;
  skos:narrower psr:-RS9D4FLP-X .

psr:-SKGJ9CKK-N
  skos:prefLabel "géométrie algébrique"@fr, "algebraic geometry"@en ;
  a skos:Concept ;
  skos:narrower psr:-RS9D4FLP-X .

psr:-LMDZ11CG-L
  skos:prefLabel "algèbre homotopique"@fr, "homotopical algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-RS9D4FLP-X .

psr: a skos:ConceptScheme .