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.

In dependent type theory, one can also use an arbitrary type $I$ instead of a finite type to define functions of arbitrary arity:

$$\overline{x}:\prod_{x:I} A(x) \vdash f(\overline{x}):B$$

An operation of arbitrary arity is a function where the sources and target are the same:

$$\overline{x}:I \to A \vdash f(\overline{x}):A$$

These are important for defining impredicative structures such as suplattices, frames, complete Heyting algebras, and Grothendieck topoi.

Moreover, in addition to functions and operations of arbitrary arity, one also has type families and dependent functions of arbitrary arity. These are, in essence, families indexed by dependent product types:

$$\overline{x}:\prod_{x:I} A(x) \vdash B(\overline{x}) \; \mathrm{type} \qquad \overline{x}:\prod_{x:I} A(x) \vdash f(\overline{x}):B(\overline{x})$$

These are important for defining identity types $\mathrm{Id}_{I, A}(\overline{x})$, indexed heterogeneous identity types $\mathrm{hId}_{I, A, B}(\overline{x}, p, \overline{y})$, and bridge types $\mathrm{Br}_{I, A}(\overline{x})$ of arbitrary arity, as well as expressing function application to identifications for functions of arbitrary arity.

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

• commutative operation

References

• Wikipedia, n-ary group

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