---
tag: WFVC137TYB
kind: concept
name: Infinitary operation
msc: [08A65]
sources:
  - url: https://en.wikipedia.org/wiki/Arity
---

## Summary

An infinitary operation on a set $A$ takes an infinite family of elements of $A$ and returns an element of $A$.
Its [[arity|MSXX2P86CY]] is an infinite cardinal or ordinal, in contrast to a [[finitary operation|EJ8YJVM9BW]].
The supremum of an arbitrary subset of a complete lattice and countable unions in a $\sigma$-algebra of sets are typical infinitary operations.
Algebras with infinitary operations need care, because some results of finitary [[universal algebra|W82R6N0GM9]] fail for them.

## Explanations

### L1 human

Most operations take two or three inputs, but some take infinitely many.
In the interval $[0, 1]$, every sequence $x_0, x_1, x_2, \dots$ has a least upper bound, which is again in $[0, 1]$.
So "take the supremum of a sequence" is an operation with countably many inputs.
In contrast, "add up a sequence" is not an operation on the real numbers, because many series do not converge.