nLab arity

N-ary operations

Context

Algebra

higher algebra

universal algebra

N-ary operations

Definitions

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

$\phi \;\colon\; \big( \prod_{i:[n]} S \big) \,=\, S^n \overset{\;\;\;\;\;}{\longrightarrow} S$

from the $n$th cartesian power $S^n$ of $S$ to $S$ itself, where [n] is a set with $n$ elements. The arity of the operation is $n$.

More generally, an n-ary operation in a multicategory is just a multimorphism.

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.