nLab
arity

N-ary operations

N-ary operations

Definitions

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

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

from the nnth cartesian power S nS^n of SS to SS itself, where [n] is a set with nn elements. The arity of the operation is nn.

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

Properties

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

Last revised on May 7, 2021 at 05:45:28. See the history of this page for a list of all contributions to it.