Abbreviation: CanSgrp

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: $x\cdot z=y\cdot z\Longrightarrow x=y$

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:

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

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

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

Cancellative monoids

Semigroups