Abbreviation: CanSgrp

A \emph{left cancellative semigroup} is a semigroup $\mathbf{S}=\langle S,\cdot \rangle $ such that

$\cdot $ is left cancellative: $z\cdot x=z\cdot y\Longrightarrow x=y$

Let $\mathbf{S}$ and $\mathbf{T}$ be left 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.

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

Left cancellative monoids

Cancellative semigroups

Semigroups