---
tag: 3M724K3B50
kind: concept
name: Inverse semigroup
msc: [20M18]
specializes: [XGJKF6T7FZ]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_semigroup
same_as:
  wikidata: Q3022118
  wikipedia: Inverse semigroup
---

## Summary

An inverse semigroup is a [[semigroup|0XMEM406JM]] in which each element $a$ has exactly one element $a^{-1}$ with $a a^{-1} a = a$ and $a^{-1} a a^{-1} = a^{-1}$.
The model example is the semigroup of partial bijections of a set, just as the model group is a group of permutations.
By the Wagner–Preston theorem, every inverse semigroup embeds in a semigroup of partial bijections.
Inverse semigroups describe partial symmetries, and they appear in the theory of tilings and of operator algebras.

## Explanations

### L1 human

A group collects symmetries that move a whole object onto itself.
Many structures have only partial symmetries, which match one part of the object with another part.
Partial symmetries can be composed where they fit and reversed, and they form an inverse semigroup.
For example, the map that sends $1$ to $2$ and is undefined elsewhere is a partial bijection of $\set{1, 2, 3}$, and its inverse sends $2$ to $1$.