nLab arity

N-ary operations

N-ary operations


Given a natural number nn, an n-ary operation on a set SS is a function

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

from the function set Fin(n)S\mathrm{Fin}(n) \to S to SS itself, where Fin(n)\mathrm{Fin}(n) is the finite set with nn elements.

The arity of the operation is nn.

In general, if the natural number nn 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.


Every set SS with an nn-ary operation ϕ\phi comes with an endomorphism called the nn-th power operation

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

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

See also

nn-ary algebraic structures


Last revised on May 12, 2024 at 20:53:44. See the history of this page for a list of all contributions to it.