Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeMay 5th 2021
   • PermaLink
   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

2. • CommentRowNumber2.
   • CommentAuthorDavid_Corfield
   • CommentTimeMay 6th 2021
   • PermaLink
   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 $n$, 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,

3. • CommentRowNumber3.
   • CommentAuthorvarkor
   • CommentTimeMay 6th 2021
   • PermaLink
   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 $\prod_{i:[n]} (-)_i \colon S^n \to S$? Am I being slow?

   I am also confused. An n-ary operation on a set $S$ is just a function $S^n \to S$.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMay 6th 2021
   • PermaLink
   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 $S^n$ or $\underset{i \in [n]}{\prod} S$.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMay 7th 2021
   • PermaLink
   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

6. • CommentRowNumber6.
   • CommentAuthornLab edit announcer
   • CommentTimeDec 9th 2022
   • PermaLink
   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

7. • CommentRowNumber7.
   • CommentAuthornLab edit announcer
   • CommentTimeDec 9th 2022
   • PermaLink
   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

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJan 28th 2024
   • PermaLink
   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