---
tag: XD10P9CJMD
kind: concept
name: Unary algebra
msc: [08A60]
specializes: [QJGPT2W0FT]
sources:
  - url: https://en.wikipedia.org/wiki/Unary_operation
---

## Summary

A unary algebra is a set with a family of [[unary operations|Q2W26ZY10B]], maps from the set to itself, and no operations of other arities.
It is the simplest kind of [[algebraic structure|QJGPT2W0FT]], and yet it already contains the theory of functional graphs and of automata.
A unary algebra with operations indexed by a set $I$ is the same as an action of the free monoid on $I$.
A unary algebra with one operation is called a mono-unary algebra.

## Explanations

### L1 human

Take a set and some ways of moving each element to another element.
That is a unary algebra.
With one operation $f$, draw an arrow from each $x$ to $f(x)$, so that every point has exactly one outgoing arrow.
For a finite set, the picture is made of cycles with trees hanging off them.
For example, the integers with the successor map $n \mapsto n + 1$ form a unary algebra whose picture is one infinite line.
A [[semiautomaton|RTMYXRDGWR]] is a unary algebra with one operation for each input letter.