Cancellative semigroups

Abbreviation: CanSgrp

Definition

A 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.

Properties

Classtype quasivariety no no no no no no no no no no

Finite members

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