---
tag: 657D5Q37J1
kind: definition
name: Inverse semigroup
msc: [20M18]
defines: 3M724K3B50
uses: [0XMEM406JM, XGJKF6T7FZ, 0N5NJZKKN3]
sources:
  - url: https://en.wikipedia.org/wiki/Inverse_semigroup
---

## Definition

Let $S$ be a [[semigroup|0XMEM406JM]].
The semigroup $S$ is an *inverse semigroup* if for every $a \in S$ there is exactly one $b \in S$ with
$$aba = a \quad \text{and} \quad bab = b.$$
This $b$ is written $a^{-1}$.

## Remarks

Equivalently, $S$ is [[regular|XGJKF6T7FZ]] and any two [[idempotents|0N5NJZKKN3]] of $S$ commute.
In an inverse semigroup, $(a^{-1})^{-1} = a$ and $(ab)^{-1} = b^{-1} a^{-1}$.

Example: the symmetric inverse monoid $I_X$ on a set $X$, the set of all bijections $f \colon A \to B$ between subsets $A, B \subseteq X$.
The product $fg$ is the composite $f \circ g$, defined on the points $x$ with $x$ in the domain of $g$ and $g(x)$ in the domain of $f$.
For $f \colon A \to B$, the inverse bijection $f^{-1} \colon B \to A$ satisfies $f f^{-1} f = f$ and $f^{-1} f f^{-1} = f^{-1}$, and it is the only element that does.
Every group is also an inverse semigroup.

Non-example: the left zero semigroup $\set{e, f}$ with $xy = x$ for all $x, y$.
Here $exe = e$ and $xex = x$ for both $x = e$ and $x = f$, so $e$ has two such partners and uniqueness fails.
Accordingly, the idempotents $e$ and $f$ do not commute: $ef = e$ but $fe = f$.