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nLab > Latest Changes: arity

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- CommentRowNumber1.
- CommentAuthornLab edit announcer
- CommentTimeMay 5th 2021
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Author: nLab edit announcer
Format: MarkdownItexdefining arity of a function Anonymous <a href="https://ncatlab.org/nlab/revision/arity/1">v1</a>, <a href="https://ncatlab.org/nlab/show/arity">current</a>

defining arity of a function

Anonymous

v1, current
- CommentRowNumber2.
- CommentAuthorDavid_Corfield
- CommentTimeMay 6th 2021
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Author: David_Corfield
Format: MarkdownItex> Given a [[cardinal number]] $n$, an __n-ary operation__ on a [[set]] $S$ is a [[function]] $\prod_{i:[n]} (-)_i \colon S^n \to S$ from the [[cartesian power]] $S^n$ to $S$, where [n] is a set with $n$ elements. The __arity__ of the operation is $n$. How are we to read $\prod_{i:[n]} (-)_i \colon S^n \to S$? Am I being slow? Also, we need 'arity' for relations. E.g., at [[signature (in logic)]] we have > A set $Rel(\Sigma)$ whose elements are called **relation symbols**, equipped with a function $ar: Rel(\Sigma) \to S^*$ to the [[free monoid]] on $S$ which prescribes an [[arity]] for each relation symbol,

Given a cardinal number $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ , an n-ary operation on a set $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>$ is a function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo lspace="0.16667em" rspace="0.16667em">∏</mo> <mrow><mi>i</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo lspace="0.11111em">:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\prod_{i:[n]} (-)_i \colon S^n \to S</annotation></semantics></math>$ from the cartesian power $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>$ to $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>$ , where [n] is a set with $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ elements. The arity of the operation is $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>$ .

How are we to read $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo lspace="0.16667em" rspace="0.16667em">∏</mo> <mrow><mi>i</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo lspace="0.11111em">:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\prod_{i:[n]} (-)_i \colon S^n \to S</annotation></semantics></math>$ ? Am I being slow?

Also, we need ’arity’ for relations. E.g., at signature (in logic) we have

A set $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>Rel</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rel(\Sigma)</annotation></semantics></math>$ whose elements are called relation symbols, equipped with a function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ar</mi><mo>:</mo><mi>Rel</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>S</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">ar: Rel(\Sigma) \to S^*</annotation></semantics></math>$ to the free monoid on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>$ which prescribes an arity for each relation symbol,
- CommentRowNumber3.
- CommentAuthorvarkor
- CommentTimeMay 6th 2021
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Author: varkor
Format: MarkdownItex> How are we to read $\prod_{i:[n]} (-)_i \colon S^n \to S$? Am I being slow? I am also confused. An n-ary operation on a set $S$ is just a function $S^n \to S$.

How are we to read $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mo lspace="0.16667em" rspace="0.16667em">∏</mo> <mrow><mi>i</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="0.11111em" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo lspace="0.11111em">:</mo><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\prod_{i:[n]} (-)_i \colon S^n \to S</annotation></semantics></math>$ ? Am I being slow?

I am also confused. An n-ary operation on a set $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>$ is just a function $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">S^n \to S</annotation></semantics></math>$ .
- CommentRowNumber4.
- CommentAuthorUrs
- CommentTimeMay 6th 2021
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Author: Urs
Format: MarkdownItexI guess it's just a typo coming from a change of mind between writing $S^n$ or $\underset{i \in [n]}{\prod} S$.

I guess it’s just a typo coming from a change of mind between writing $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">S^n</annotation></semantics></math>$ or $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><munder><mo lspace="0.16667em" rspace="0.16667em">∏</mo><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow></munder><mi>S</mi></mrow><annotation encoding="application/x-tex">\underset{i \in [n]}{\prod} S</annotation></semantics></math>$ .
- CommentRowNumber5.
- CommentAuthorUrs
- CommentTimeMay 7th 2021
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Author: Urs
Format: MarkdownItexI have fixed it. Also in the Properties-section further below. <a href="https://ncatlab.org/nlab/revision/diff/arity/2">diff</a>, <a href="https://ncatlab.org/nlab/revision/arity/2">v2</a>, <a href="https://ncatlab.org/nlab/show/arity">current</a>

I have fixed it. Also in the Properties-section further below.

diff, v2, current
- CommentRowNumber6.
- CommentAuthornLab edit announcer
- CommentTimeDec 9th 2022
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Author: nLab edit announcer
Format: MarkdownItexlinking to [[operad]] Anonymous <a href="https://ncatlab.org/nlab/revision/diff/arity/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/arity/5">v5</a>, <a href="https://ncatlab.org/nlab/show/arity">current</a>

linking to operad

Anonymous

diff, v5, current
- CommentRowNumber7.
- CommentAuthornLab edit announcer
- CommentTimeDec 9th 2022
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Author: nLab edit announcer
Format: MarkdownItexadded section on finitary operations Anonymous <a href="https://ncatlab.org/nlab/revision/diff/arity/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/arity/6">v6</a>, <a href="https://ncatlab.org/nlab/show/arity">current</a>

added section on finitary operations

Anonymous

diff, v6, current
- CommentRowNumber8.
- CommentAuthorUrs
- CommentTimeJan 28th 2024
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Author: Urs
Format: MarkdownItexhave slightly expanded at the point where the term "finitary groupoid" appears, for clarification. <a href="https://ncatlab.org/nlab/revision/diff/arity/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/arity/7">v7</a>, <a href="https://ncatlab.org/nlab/show/arity">current</a>

have slightly expanded at the point where the term “finitary groupoid” appears, for clarification.

diff, v7, current

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nLab > Latest Changes: arity