=====Commutative semigroups=====

Abbreviation: **CSgrp**
====Definition====
A \emph{commutative semigroup} is a [[semigroups]] $\mathbf{S}=\langle
S,\cdot \rangle $ such that

$\cdot $ is commutative:  $xy=yx$
====Definition====
A \emph{commutative semigroup} is a structure $\mathbf{S}=\langle
S,\cdot \rangle $, where $\cdot $ is an infix binary operation, called
the \emph{semigroup product}, such that


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


$\cdot $ is commutative:  $xy=yx$
==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be commutative semigroups. A morphism from 
$\mathbf{S}$ to $\mathbf{T}$ is a function $h:Sarrow T$ that is a
homomorphism: 

$h(xy)=h(x)h(y)$

====Examples====
Example 1: $\langle \mathbb{N},+\rangle $, the natural numbers, with additition.



====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable in polynomial time |
^[[Quasiequational theory]]  |decidable |
^[[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]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  |no |
^[[Strong amalgamation property]]  |no |
^[[Epimorphisms are surjective]]  |no |
====Finite members====

$\begin{array}{lr}
[[Search for finite commutative semigroups]]

f(1)= &1\\
f(2)= &3\\
f(3)= &12\\
f(4)= &58\\
f(5)= &325\\
f(6)= &2143\\
f(7)= &17291\\
\end{array}$

====Subclasses====
[[Semilattices]] 

[[Commutative monoids]] 

====Superclasses====
[[Semigroups]] 

[[Partial commutative semigroups]] 


====References====

[(Ln19xx>
)]