nLab arity

Definitions

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

$$\phi \;\colon\; \big( \mathrm{Fin}(n) \to S \big) \longrightarrow S$$

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 equipped with finitary operations are also called finitary magmas (or “finitary groupoids” in older terminology which now clashes with another meaning of groupoid, see at historical notes on quasigroups).

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.

See also

• magma

• monoid

• operation

• multicategory

• operad

$n$-ary algebraic structures

• n-ary semigroup

• n-ary quasigroup

• n-ary group

• group

References

• Wikipedia, n-ary group

• Steven Duplij, Polyadic Algebraic Structures, IOP (2022) [ISBN:978-0-7503-2646-9]