---
tag: 9HPF7TE18C
kind: concept
name: Nullary operation
msc: [08A62, 08A02]
specializes: [EJ8YJVM9BW]
sources:
  - url: https://en.wikipedia.org/wiki/Arity
---

## Summary

A nullary operation on a set $A$ is a [[finitary operation|EJ8YJVM9BW]] of [[arity|MSXX2P86CY]] $0$, that is, a map from the one-element set $A^0$ to $A$.
Such a map is the same as the choice of one element of $A$, so nullary operations are the constants of an [[algebraic structure|QJGPT2W0FT]].
Treating constants as operations lets identity elements, such as $0$ in a ring or $e$ in a [[group|G5S1M9]], be part of the signature.

## Explanations

### L1 human

An operation with no inputs cannot depend on anything, so it always gives the same element.
That element is a constant.
For example, in the integers with addition, the number $0$ can be seen as a nullary operation.
When $0$ is part of the signature, a [[homomorphism|KBXSZC6BV1]] must send the constant $0$ to the constant $0$ of the target.