=====Cancellative semigroups=====

Abbreviation: **CanSgrp**
====Definition====
A \emph{cancellative semigroup} is a [[semigroup]] $\mathbf{S}=\langle
S,\cdot\rangle$ 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 $\mathbf{S}$ and $\mathbf{T}$ be cancellative semigroups. A morphism from 
$\mathbf{S}$ to $\mathbf{T}$ is a function $h:S\rightarrow T$ that is a
homomorphism: 

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

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


====Basic results====


====Properties====
^[[Classtype]]  |quasivariety |
^[[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]]  | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  |no |
^[[Strong amalgamation property]]  |no |
^[[Epimorphisms are surjective]]  |no |

====Finite members====

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


====Subclasses====
[[Cancellative monoids]] 

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


====References====

[(Ln19xx>
)]