cancellative_commutative_monoids

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=====Cancellative commutative monoids=====

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

$\cdot $ is commutative:  $x\cdot y=y\cdot x$
==Morphisms==
Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative commutative monoids. A morphism from $\mathbf{M}$
to $\mathbf{N}$ is a function $h:Marrow N$ 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. 


====Basic results====
All commutative free monoids are cancellative.

All finite commutative (left or right) cancellative monoids are reducts of [[abelian groups]].

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

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


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

[[Cancellative commutative residuated lattices]] 

====Superclasses====
[[Cancellative commutative semigroups]] 

[[Cancellative monoids]] 

[[Commutative monoids]] 


====References====

[(Ln19xx>
)]

Trace: • action_lattices • lattices • symmetric_relations • rectangular_bands • integral_domains • locally_finite • distributive_lattices

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