---
tag: EJ8YJVM9BW
kind: concept
name: Finitary operation
msc: [08A62, 08A02]
sources:
  - url: https://en.wikipedia.org/wiki/Operation_(mathematics)
---

## Summary

A finitary operation on a set $A$ takes $n$ elements of $A$ and returns an element of $A$, for a fixed natural number $n$.
The number $n$ is its [[arity|MSXX2P86CY]].
[[Binary operations|0M418FE7BB]] such as addition are the most common case, but [[unary|Q2W26ZY10B]], [[nullary|9HPF7TE18C]] and [[ternary|WC47MFR6GW]] operations also occur.
[[Algebraic structures|QJGPT2W0FT]] are sets equipped with finitary operations.

## Explanations

### L1 human

An operation is a rule that turns some elements of a set into one element of the same set.
"Finitary" means that the number of inputs is a fixed natural number.
On the integers, $x \mapsto -x$ uses one input, $(x, y) \mapsto x + y$ uses two, and $(x, y, z) \mapsto x - y + z$ uses three.
The result must stay inside the set: subtraction is an operation on the integers but not on the natural numbers.
Operations with infinitely many inputs are called [[infinitary|WFVC137TYB]].