cancellative_commutative_semigroups

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=====Cancellative commutative semigroups=====

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

$\cdot $ is \emph{cancellative}:  $x\cdot z=y\cdot z\Longrightarrow x=y$
==Morphisms==
Let $\mathbf{S}$ and $\mathbf{T}$ be cancellative commutative 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 commutative monoids]] 

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

[[Commutative semigroups]] 


====References====

[(Ln19xx>
)]

Trace: • closure_algebras • kleene_algebras • distributive_lattice_expansions • halmos_algebras

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