@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:-DZBL40KV-9
  skos:prefLabel "hyperoperation"@en, "hyperopération"@fr ;
  skos:narrower psr:-M3CF2352-9, psr:-DND5DXXP-B ;
  dc:created "2023-08-23"^^xsd:date ;
  skos:broader psr:-S20PBJCF-M, psr:-BSS33T4W-C ;
  skos:definition """In mathematics, the <b>hyperoperation sequence</b> is an infinite sequence of arithmetic operations (called <i>hyperoperations</i> in this context) that starts with a unary operation (the successor function with <i>n</i> = 0). The sequence continues with the binary operations of addition (<i>n</i> = 1), multiplication (<i>n</i> = 2), and exponentiation (<i>n</i> = 3). After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity.  For the operations beyond exponentiation, the <i>n</i>th member of this sequence is named by Reuben Goodstein after the Greek prefix of <i>n</i> suffixed with <i>-ation</i> (such as tetration (<i>n</i> = 4), pentation (<i>n</i> = 5), hexation (<i>n</i> = 6), etc.)  and can be written as using <i>n</i> − 2 arrows in Knuth's up-arrow notation. Each hyperoperation may be understood recursively in terms of the previous one by:  <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 ab=\\\\underbrace {a[n-1](a[n-1](a[n-1](\\\\cdots [n-1](a[n-1](a[n-1]a))\\\\cdots )))} _{\\\\displaystyle b{\\\\mbox{ copies of }}a},\\\\quad n\\\\geq 2}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <mo stretchy="false">[</mo>         <mi>n</mi>         <mo stretchy="false">]</mo>         <mi>b</mi>         <mo>=</mo>         <munder>           <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">             <munder>               <mrow>                 <mi>a</mi>                 <mo stretchy="false">[</mo>                 <mi>n</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>                 <mo stretchy="false">]</mo>                 <mo stretchy="false">(</mo>                 <mi>a</mi>                 <mo stretchy="false">[</mo>                 <mi>n</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>                 <mo stretchy="false">]</mo>                 <mo stretchy="false">(</mo>                 <mi>a</mi>                 <mo stretchy="false">[</mo>                 <mi>n</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>                 <mo stretchy="false">]</mo>                 <mo stretchy="false">(</mo>                 <mo>⋯<!-- ⋯ --></mo>                 <mo stretchy="false">[</mo>                 <mi>n</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>                 <mo stretchy="false">]</mo>                 <mo stretchy="false">(</mo>                 <mi>a</mi>                 <mo stretchy="false">[</mo>                 <mi>n</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>                 <mo stretchy="false">]</mo>                 <mo stretchy="false">(</mo>                 <mi>a</mi>                 <mo stretchy="false">[</mo>                 <mi>n</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>                 <mo stretchy="false">]</mo>                 <mi>a</mi>                 <mo stretchy="false">)</mo>                 <mo stretchy="false">)</mo>                 <mo>⋯<!-- ⋯ --></mo>                 <mo stretchy="false">)</mo>                 <mo stretchy="false">)</mo>                 <mo stretchy="false">)</mo>               </mrow>               <mo>⏟<!-- ⏟ --></mo>             </munder>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mstyle displaystyle="true" scriptlevel="0">               <mi>b</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mstyle displaystyle="false" scriptlevel="0">                   <mtext> copies of </mtext>                 </mstyle>               </mrow>               <mi>a</mi>             </mstyle>           </mrow>         </munder>         <mo>,</mo>         <mspace width="1em"></mspace>         <mi>n</mi>         <mo>≥<!-- ≥ --></mo>         <mn>2</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ab=\\\\underbrace {a[n-1](a[n-1](a[n-1](\\\\cdots [n-1](a[n-1](a[n-1]a))\\\\cdots )))} _{\\\\displaystyle b{\\\\mbox{ copies of }}a},\\\\quad n\\\\geq 2}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5b37b38e6798ec67d72e429a051aa7641fb571" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:79.82ex; height:7.176ex;" alt="{\\\\displaystyle ab=\\\\underbrace {a[n-1](a[n-1](a[n-1](\\\\cdots [n-1](a[n-1](a[n-1]a))\\\\cdots )))} _{\\\\displaystyle b{\\\\mbox{ copies of }}a},\\\\quad n\\\\geq 2}"></span></dd></dl> It may also be defined according to the recursion rule part of the definition, as in Knuth's up-arrow version of the Ackermann function:  <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 ab=a[n-1]\\\\left(a\\\\left(b-1\\
ight)\\
ight),\\\\quad n\\\\geq 1}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <mo stretchy="false">[</mo>         <mi>n</mi>         <mo stretchy="false">]</mo>         <mi>b</mi>         <mo>=</mo>         <mi>a</mi>         <mo stretchy="false">[</mo>         <mi>n</mi>         <mo>−<!-- − --></mo>         <mn>1</mn>         <mo stretchy="false">]</mo>         <mrow>           <mo>(</mo>           <mrow>             <mi>a</mi>             <mo stretchy="false">[</mo>             <mi>n</mi>             <mo stretchy="false">]</mo>             <mrow>               <mo>(</mo>               <mrow>                 <mi>b</mi>                 <mo>−<!-- − --></mo>                 <mn>1</mn>               </mrow>               <mo>)</mo>             </mrow>           </mrow>           <mo>)</mo>         </mrow>         <mo>,</mo>         <mspace width="1em"></mspace>         <mi>n</mi>         <mo>≥<!-- ≥ --></mo>         <mn>1</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle ab=a[n-1]\\\\left(a\\\\left(b-1\\
ight)\\
ight),\\\\quad n\\\\geq 1}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acfe8d681527fde64d3bcacced77270f96900f25" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.646ex; height:2.843ex;" alt="{\\\\displaystyle ab=a[n-1]\\\\left(a\\\\left(b-1\\
ight)\\
ight),\\\\quad n\\\\geq 1}"></span></dd></dl> This can be used to easily show numbers much larger than those which scientific notation can, such as Skewes's number and googolplexplex (e.g. <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 5050}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mn>50</mn>         <mo stretchy="false">[</mo>         <mn>50</mn>         <mo stretchy="false">]</mo>         <mn>50</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle 5050}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e849d9d2bf3c592596ad3b7670cb605f68ccf252" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.268ex; height:2.843ex;" alt="{\\\\displaystyle 5050}"></span> is much larger than Skewes's number and googolplexplex), but there are some numbers which even they cannot easily show, such as Graham's number and TREE(3). This recursion rule is common to many variants of hyperoperations. 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Hyperoperation">https://en.wikipedia.org/wiki/Hyperoperation</a>)"""@en, """En mathématiques, les <b>hyperopérations</b>  (ou <b>hyperopérateurs</b>) constituent une suite infinie d'opérations qui prolonge logiquement la suite des opérations arithmétiques élémentaires suivantes :  <ol><li>addition (n = 1) : <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 {{H_{1}(a,b)=a+b} \\\\atop \\\\,}{= \\\\atop \\\\,}{a\\\\,+ \\\\atop \\\\,}{{\\\\underbrace {1+1+\\\\cdots +1} } \\\\atop b{\\	ext{ termes}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow class="MJX-TeXAtom-ORD">               <msub>                 <mi>H</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>1</mn>                 </mrow>               </msub>               <mo stretchy="false">(</mo>               <mi>a</mi>               <mo>,</mo>               <mi>b</mi>               <mo stretchy="false">)</mo>               <mo>=</mo>               <mi>a</mi>               <mo>+</mo>               <mi>b</mi>             </mrow>             <mspace width="thinmathspace"></mspace>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mo>=</mo>             <mspace width="thinmathspace"></mspace>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow>               <mi>a</mi>               <mspace width="thinmathspace"></mspace>               <mo>+</mo>             </mrow>             <mspace width="thinmathspace"></mspace>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow class="MJX-TeXAtom-ORD">               <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">                 <munder>                   <mrow>                     <mn>1</mn>                     <mo>+</mo>                     <mn>1</mn>                     <mo>+</mo>                     <mo>⋯<!-- ⋯ --></mo>                     <mo>+</mo>                     <mn>1</mn>                   </mrow>                   <mo>⏟<!-- ⏟ --></mo>                 </munder>               </mrow>             </mrow>             <mrow>               <mi>b</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mtext> termes</mtext>               </mrow>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {{H_{1}(a,b)=a+b} \\\\atop \\\\,}{= \\\\atop \\\\,}{a\\\\,+ \\\\atop \\\\,}{{\\\\underbrace {1+1+\\\\cdots +1} } \\\\atop b{\\	ext{ termes}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8076bdbf26ffb05fbe8d67b41d0ff857b08471" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.417ex; height:6.509ex;" alt="{\\\\displaystyle {{H_{1}(a,b)=a+b} \\\\atop \\\\,}{= \\\\atop \\\\,}{a\\\\,+ \\\\atop \\\\,}{{\\\\underbrace {1+1+\\\\cdots +1} } \\\\atop b{\\	ext{ termes}}}}"></span></dd></dl></li> <li>multiplication (n = 2) : <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 {{H_{2}(a,b)=a\\	imes b=\\\\ } \\\\atop {\\\\ }}{{\\\\underbrace {a+a+\\\\cdots +a} } \\\\atop b{\\	ext{ termes}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow class="MJX-TeXAtom-ORD">               <msub>                 <mi>H</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>2</mn>                 </mrow>               </msub>               <mo stretchy="false">(</mo>               <mi>a</mi>               <mo>,</mo>               <mi>b</mi>               <mo stretchy="false">)</mo>               <mo>=</mo>               <mi>a</mi>               <mo>×<!-- × --></mo>               <mi>b</mi>               <mo>=</mo>               <mtext> </mtext>             </mrow>             <mrow class="MJX-TeXAtom-ORD">               <mtext> </mtext>             </mrow>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow class="MJX-TeXAtom-ORD">               <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">                 <munder>                   <mrow>                     <mi>a</mi>                     <mo>+</mo>                     <mi>a</mi>                     <mo>+</mo>                     <mo>⋯<!-- ⋯ --></mo>                     <mo>+</mo>                     <mi>a</mi>                   </mrow>                   <mo>⏟<!-- ⏟ --></mo>                 </munder>               </mrow>             </mrow>             <mrow>               <mi>b</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mtext> termes</mtext>               </mrow>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {{H_{2}(a,b)=a\\	imes b=\\\\ } \\\\atop {\\\\ }}{{\\\\underbrace {a+a+\\\\cdots +a} } \\\\atop b{\\	ext{ termes}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a8e05c4c3a2139b15a24dc8855e940d4d1a1ad8" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:35.95ex; height:6.343ex;" alt="{\\\\displaystyle {{H_{2}(a,b)=a\\	imes b=\\\\ } \\\\atop {\\\\ }}{{\\\\underbrace {a+a+\\\\cdots +a} } \\\\atop b{\\	ext{ termes}}}}"></span></dd></dl></li> <li>exponentiation (n = 3) : <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 {{H_{3}(a,b)=a^{b}=\\\\ } \\\\atop {\\\\ }}{{\\\\underbrace {a\\	imes a\\	imes \\\\cdots \\	imes a} } \\\\atop b{\\	ext{ facteurs}}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow class="MJX-TeXAtom-ORD">               <msub>                 <mi>H</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mn>3</mn>                 </mrow>               </msub>               <mo stretchy="false">(</mo>               <mi>a</mi>               <mo>,</mo>               <mi>b</mi>               <mo stretchy="false">)</mo>               <mo>=</mo>               <msup>                 <mi>a</mi>                 <mrow class="MJX-TeXAtom-ORD">                   <mi>b</mi>                 </mrow>               </msup>               <mo>=</mo>               <mtext> </mtext>             </mrow>             <mrow class="MJX-TeXAtom-ORD">               <mtext> </mtext>             </mrow>           </mfrac>         </mrow>         <mrow class="MJX-TeXAtom-ORD">           <mfrac linethickness="0">             <mrow class="MJX-TeXAtom-ORD">               <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">                 <munder>                   <mrow>                     <mi>a</mi>                     <mo>×<!-- × --></mo>                     <mi>a</mi>                     <mo>×<!-- × --></mo>                     <mo>⋯<!-- ⋯ --></mo>                     <mo>×<!-- × --></mo>                     <mi>a</mi>                   </mrow>                   <mo>⏟<!-- ⏟ --></mo>                 </munder>               </mrow>             </mrow>             <mrow>               <mi>b</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mtext> facteurs</mtext>               </mrow>             </mrow>           </mfrac>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {{H_{3}(a,b)=a^{b}=\\\\ } \\\\atop {\\\\ }}{{\\\\underbrace {a\\	imes a\\	imes \\\\cdots \\	imes a} } \\\\atop b{\\	ext{ facteurs}}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e8cf072307bec767aa75573d5fcc5cd15ab53d8a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.05ex; height:6.343ex;" alt="{\\\\displaystyle {{H_{3}(a,b)=a^{b}=\\\\ } \\\\atop {\\\\ }}{{\\\\underbrace {a\\	imes a\\	imes \\\\cdots \\	imes a} } \\\\atop b{\\	ext{ facteurs}}}}"></span></dd></dl></li></ol> Reuben Goodstein</span> proposa de baptiser les opérations au-delà de l'exponentiation en utilisant des préfixes grecs : <i>tétration</i> (<i>n</i> = 4), <i>pentation</i> (<i>n</i> = 5), hexation (<i>n</i> = 6), etc. L'hyperopération à l'ordre <i>n</i> peut se noter à l'aide d'une flèche de Knuth au rang <i>n</i> – 2. <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 H_{n}(a,b)=a\\\\uparrow ^{n-2}b}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <msub>           <mi>H</mi>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>           </mrow>         </msub>         <mo stretchy="false">(</mo>         <mi>a</mi>         <mo>,</mo>         <mi>b</mi>         <mo stretchy="false">)</mo>         <mo>=</mo>         <mi>a</mi>         <msup>           <mo stretchy="false">↑<!-- ↑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>n</mi>             <mo>−<!-- − --></mo>             <mn>2</mn>           </mrow>         </msup>         <mi>b</mi>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle H_{n}(a,b)=a\\\\uparrow ^{n-2}b}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a98819c84b8b44bc11a731cc6a4f5111e16c39" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.318ex; height:3.176ex;" alt="{\\\\displaystyle H_{n}(a,b)=a\\\\uparrow ^{n-2}b}"></span>. La flêche de Knuth au rang <i>m</i> est définie récursivement par : <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\\\\uparrow ^{-1}b=a+b\\\\,}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <msup>           <mo stretchy="false">↑<!-- ↑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msup>         <mi>b</mi>         <mo>=</mo>         <mi>a</mi>         <mo>+</mo>         <mi>b</mi>         <mspace width="thinmathspace"></mspace>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\uparrow ^{-1}b=a+b\\\\,}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e80f9c88cf4f3fc89ca85a6faea36dc107d7743" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.566ex; height:3.009ex;" alt="{\\\\displaystyle a\\\\uparrow ^{-1}b=a+b\\\\,}"></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 a\\\\uparrow ^{m}b=\\\\underbrace {a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m-1}\\\\left[a\\\\uparrow ^{m-1}\\\\left(\\\\ldots \\\\left[a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m-1}a\\
ight)\\
ight]\\\\ldots \\
ight)\\
ight]\\
ight)} _{\\\\displaystyle b{\\\\mbox{ copies de }}a},\\\\quad m\\\\geq 0}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <msup>           <mo stretchy="false">↑<!-- ↑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>           </mrow>         </msup>         <mi>b</mi>         <mo>=</mo>         <munder>           <mrow class="MJX-TeXAtom-OP MJX-fixedlimits">             <munder>               <mrow>                 <mi>a</mi>                 <msup>                   <mo stretchy="false">↑<!-- ↑ --></mo>                   <mrow class="MJX-TeXAtom-ORD">                     <mi>m</mi>                     <mo>−<!-- − --></mo>                     <mn>1</mn>                   </mrow>                 </msup>                 <mrow>                   <mo>(</mo>                   <mrow>                     <mi>a</mi>                     <msup>                       <mo stretchy="false">↑<!-- ↑ --></mo>                       <mrow class="MJX-TeXAtom-ORD">                         <mi>m</mi>                         <mo>−<!-- − --></mo>                         <mn>1</mn>                       </mrow>                     </msup>                     <mrow>                       <mo>[</mo>                       <mrow>                         <mi>a</mi>                         <msup>                           <mo stretchy="false">↑<!-- ↑ --></mo>                           <mrow class="MJX-TeXAtom-ORD">                             <mi>m</mi>                             <mo>−<!-- − --></mo>                             <mn>1</mn>                           </mrow>                         </msup>                         <mrow>                           <mo>(</mo>                           <mrow>                             <mo>…<!-- … --></mo>                             <mrow>                               <mo>[</mo>                               <mrow>                                 <mi>a</mi>                                 <msup>                                   <mo stretchy="false">↑<!-- ↑ --></mo>                                   <mrow class="MJX-TeXAtom-ORD">                                     <mi>m</mi>                                     <mo>−<!-- − --></mo>                                     <mn>1</mn>                                   </mrow>                                 </msup>                                 <mrow>                                   <mo>(</mo>                                   <mrow>                                     <mi>a</mi>                                     <msup>                                       <mo stretchy="false">↑<!-- ↑ --></mo>                                       <mrow class="MJX-TeXAtom-ORD">                                         <mi>m</mi>                                         <mo>−<!-- − --></mo>                                         <mn>1</mn>                                       </mrow>                                     </msup>                                     <mi>a</mi>                                   </mrow>                              
<br/>     <mo>)</mo>                                 </mrow>                               </mrow>                               <mo>]</mo>                             </mrow>                             <mo>…<!-- … --></mo>                           </mrow>                           <mo>)</mo>                         </mrow>                       </mrow>                       <mo>]</mo>                     </mrow>                   </mrow>                   <mo>)</mo>                 </mrow>               </mrow>               <mo>⏟<!-- ⏟ --></mo>             </munder>           </mrow>           <mrow class="MJX-TeXAtom-ORD">             <mstyle displaystyle="true" scriptlevel="0">               <mi>b</mi>               <mrow class="MJX-TeXAtom-ORD">                 <mstyle displaystyle="false" scriptlevel="0">                   <mtext> copies de </mtext>                 </mstyle>               </mrow>               <mi>a</mi>             </mstyle>           </mrow>         </munder>         <mo>,</mo>         <mspace width="1em"></mspace>         <mi>m</mi>         <mo>≥<!-- ≥ --></mo>         <mn>0</mn>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\uparrow ^{m}b=\\\\underbrace {a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m-1}\\\\left[a\\\\uparrow ^{m-1}\\\\left(\\\\ldots \\\\left[a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m-1}a\\
ight)\\
ight]\\\\ldots \\
ight)\\
ight]\\
ight)} _{\\\\displaystyle b{\\\\mbox{ copies de }}a},\\\\quad m\\\\geq 0}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58f4a281ab56ad5844f65c66853d5c7c2ac56ba3" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:74.121ex; height:7.843ex;" alt="{\\\\displaystyle a\\\\uparrow ^{m}b=\\\\underbrace {a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m-1}\\\\left[a\\\\uparrow ^{m-1}\\\\left(\\\\ldots \\\\left[a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m-1}a\\
ight)\\
ight]\\\\ldots \\
ight)\\
ight]\\
ight)} _{\\\\displaystyle b{\\\\mbox{ copies de }}a},\\\\quad m\\\\geq 0}"></span> Elle peut aussi se définir à l'aide de la règle : <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\\\\uparrow ^{m}b=a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m}(b-1)\\
ight)}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mi>a</mi>         <msup>           <mo stretchy="false">↑<!-- ↑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>           </mrow>         </msup>         <mi>b</mi>         <mo>=</mo>         <mi>a</mi>         <msup>           <mo stretchy="false">↑<!-- ↑ --></mo>           <mrow class="MJX-TeXAtom-ORD">             <mi>m</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>           </mrow>         </msup>         <mrow>           <mo>(</mo>           <mrow>             <mi>a</mi>             <msup>               <mo stretchy="false">↑<!-- ↑ --></mo>               <mrow class="MJX-TeXAtom-ORD">                 <mi>m</mi>               </mrow>             </msup>             <mo stretchy="false">(</mo>             <mi>b</mi>             <mo>−<!-- − --></mo>             <mn>1</mn>             <mo stretchy="false">)</mo>           </mrow>           <mo>)</mo>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle a\\\\uparrow ^{m}b=a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m}(b-1)\\
ight)}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dba5e0660036bf6d69df3a8e3ea480d465778f7a" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.888ex; height:3.176ex;" alt="{\\\\displaystyle a\\\\uparrow ^{m}b=a\\\\uparrow ^{m-1}\\\\left(a\\\\uparrow ^{m}(b-1)\\
ight)}"></span>. Chacune croît plus vite que la précédente. Des suites similaires ont historiquement porté diverses appellations, telles que la <i>fonction d'Ackermann</i></span> (à 3 arguments), la <i>hiérarchie d'Ackermann</i></span>, la <i>hiérarchie de Grzegorczyk</i></span> (plus générale), la <i>version de Goodstein de la fonction d'Ackermann</i></span>, <i>hyper-n</i>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Hyperop%C3%A9ration">https://fr.wikipedia.org/wiki/Hyperop%C3%A9ration</a>)"""@fr ;
  skos:altLabel "suite d'hyperopérateurs"@fr, "hyperoperation sequence"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Hyperoperation>, <https://fr.wikipedia.org/wiki/Hyperop%C3%A9ration> ;
  dc:modified "2024-10-18"^^xsd:date ;
  a skos:Concept ;
  skos:inScheme psr: .

psr:-M3CF2352-9
  skos:prefLabel "pentation"@en, "pentation"@fr ;
  a skos:Concept ;
  skos:broader psr:-DZBL40KV-9 .

psr:-S20PBJCF-M
  skos:prefLabel "opération arithmétique"@fr, "arithmetic operation"@en ;
  a skos:Concept ;
  skos:narrower psr:-DZBL40KV-9 .

psr: a skos:ConceptScheme .
psr:-BSS33T4W-C
  skos:prefLabel "grand nombre"@fr, "large number"@en ;
  a skos:Concept ;
  skos:narrower psr:-DZBL40KV-9 .

psr:-DND5DXXP-B
  skos:prefLabel "tetration"@en, "tétration"@fr ;
  a skos:Concept ;
  skos:broader psr:-DZBL40KV-9 .