=====Cancellative monoids=====

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

$\cdot $ is left cancellative:  $z\cdot x=z\cdot y\Longrightarrow x=y$

$\cdot $ is right cancellative:  $x\cdot z=y\cdot z\Longrightarrow x=y$

==Morphisms==
Let $\mathbf{M}$ and $\mathbf{N}$ be cancellative monoids. A morphism from $\mathbf{M}$
to $\mathbf{N}$ is a function $h:M\rightarrow N$ 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 free monoids are cancellative.

All finite (left or right) cancellative monoids are reducts of [[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====

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


====Subclasses====
[[Groups]] 

[[Cancellative residuated lattices]] 

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

[[Monoids]] 


====References====

[(Ln19xx>
)]