=====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====
^[[Classtype]]  |Variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |No |
^[[Residual size]]  | |
^[[Congruence distributive]]  |No |
^[[Congruence modular]]  |No |
^[[Congruence n-permutable]]  |No |
^[[Congruence regular]]  |No |
^[[Congruence uniform]]  |No |
^[[Congruence extension property]]  |Yes |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  |No |
^[[Amalgamation property]]  |Yes |
^[[Strong amalgamation property]]  |Yes |
^[[Epimorphisms are surjective]]  |Yes |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
\end{array}$

====Subclasses====
[[Abelian groups]] 

[[Semilattices]] 

====Superclasses====
[[Inverse semigroups]] 


====References====

[(Ln19xx>
)]