=====Commutative monoids=====

Abbreviation: **CMon**
====Definition====
A \emph{commutative monoid} is a [[monoids]] $\mathbf{M}=\langle M,\cdot
,e\rangle $ such that

$\cdot $ is commutative:  $x\cdot y=y\cdot x$
====Definition====
A \emph{commutative monoid} is a structure $\mathbf{M}=\langle M,\cdot
,e\rangle $, where $\cdot $ is an infix binary operation, called the 
\emph{monoid product}, and $e$ is a constant (nullary operation), called the 
\emph{identity element}, such that


$\cdot $ is commutative:  $x\cdot y=y\cdot x$


$\cdot $ is associative:  $(x\cdot y)\cdot z=x\cdot (y\cdot z)$


$e$ is an identity for $\cdot $:  $e\cdot x=x$
==Morphisms==
Let $\mathbf{M}$ and $\mathbf{N}$ be commutative monoids. A morphism from $\mathbf{M}$
to $\mathbf{N}$ is a function $h:M\to N$ that is a homomorphism: 

$h(x\cdot y)=h(x)\cdot h(y)$, $h(e)=e$
====Examples====
Example 1: $\langle\mathbb{N},+,0\rangle$, the natural numbers, with addition and
zero. The finitely generated free commutative monoids are direct products of this one.


====Basic results====

====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  |decidable |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[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]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

$\begin{array}{lr}
f(1)= &1\\
f(2)= &2\\
f(3)= &5\\
f(4)= &19\\
f(5)= &78\\
f(6)= &421\\
f(7)= &2637\\
\end{array}$

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

[[Semilattices with identity]] 

====Superclasses====
[[Commutative semigroups]] 

[[Monoids]] 


====References====

[(Ln19xx>
)]