Table of Contents

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

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