### Table of Contents

## Commutative inverse semigroups

Abbreviation: **CInvSgrp**

### Definition

A \emph{commutative inverse semigroup} is an inverse semigroups $\mathbf{S}=\langle S,\cdot,^{-1}\rangle $ such that

$\cdot$ is commutative: $xy=yx$

### Definition

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

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

$\cdot$ is commutative: $xy=yx$

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

##### Morphisms

Let $\mathbf{S}$ and $\mathbf{T}$ be commutative 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:

### Basic results

### Properties

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &

f(3)= &

f(4)= &

f(5)= &

f(6)= &

f(7)= &

\end{array}$