## Inverse semigroups

Abbreviation: InvSgrp

### Definition

An \emph{inverse semigroup} is a structure $\mathbf{S}=\langle S,\cdot,^{-1}\rangle$ such that

$\cdot$ is associative: $(xy)z=x(yz)$

$^{-1}$ is an inverse: $xx^{-1}x=x$ and $(x^{-1})^{-1}=x$

idempotents commute: $xx^{-1}yy^{-1}=yy^{-1}xx^{-1}$

##### Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be inverse semigroups. A morphism from $\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a homomorphism:

$h(xy)=h(x)h(y)$, $h(x^{-1})=h(x)^{-1}$

### Examples

Example 1: $\langle I_X,\circ,^{-1}\rangle$, the \emph{symmetric inverse semigroup} of all one-to-one partial functions on a set $X$, with composition and function inverse. Every inverse semigroup can be embedded in a symmetric inverse semigroup.

### Basic results

$x*x=x \implies \exists y\ x=y*y^{-1}$

$\forall x\exists y\ xx^{-1}=y^{-1}y$

### Properties

Classtype Variety No No No No No No No No Yes Yes Yes

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &2 f(3)= &5 f(4)= &16 f(5)= &52 f(6)= &208 f(7)= &911 f(8)= &4637 f(9)= &26422 f(10)= &169163 f(11)= &1198651 f(12)= &9324047 f(13)= &78860687 f(14)= &719606005 f(15)= &7035514642 \end{array}$