# Start a new discussion

## Not signed in

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

## Site Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

1. defining arity of a function

Anonymous

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeMay 6th 2021

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,

• CommentRowNumber3.
• CommentAuthorvarkor
• CommentTimeMay 6th 2021

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 $S^n \to S$.

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 6th 2021

I guess it’s just a typo coming from a change of mind between writing $S^n$ or $\underset{i \in [n]}{\prod} S$.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMay 7th 2021

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