# nLab arity

N-ary operations

### Context

#### Algebra

higher algebra

universal algebra

# N-ary operations

## Definitions

Given a natural number $n$, an n-ary operation on a set $S$ is a function

$\phi:(\mathrm{Fin}(n) \to S) \to S$

from the function set $\mathrm{Fin}(n) \to S$ to $S$ itself, where $\mathrm{Fin}(n)$ is the finite set with $n$ elements. The arity of the operation is $n$. In general, if the natural number $n$ is not specified, these are called finitary operations.

Sets with finitary operations are called finitary magmas or finitary groupoids.

More generally, a finitary operation in a multicategory is just a multimorphism.

### Arbitrary arity

More generally, one could use an arbitrary set instead of a finite set. However, the generalizations are only definable in closed multicategories, rather than any multicategory.

## Properties

Every set $S$ with an $n$-ary operation $\phi$ comes with an endomorphism called the $n$-th power operation

$\array{ S & \overset{\;\;(-)^n\;\;}{\longrightarrow} & S \\ x &\mapsto& \phi \circ diag_n(x) \,, }$

where $S \overset{diag_n}{\longrightarrow} S^n$ is the diagonal morphism.