Abbreviation: CanPMon

A \emph{cancellative partial monoid} is a partial monoid such that

$\cdot$ is \emph{left-cancellative}: $x\cdot y=x\cdot z\ne *$ implies $y=z$ and

$\cdot$ is \emph{right-cancellative}: $x\cdot z=y\cdot z\ne *$ implies $x=y$.

Let $\mathbf{A}$ and $\mathbf{B}$ be cancellative partial monoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(e)=e$ and if $x\cdot y\ne *$ then $h(x \cdot y)=h(x) \cdot h(y)$.

Example 1:

See http://mathv.chapman.edu/~jipsen/uajs/CanPMon.html

$\begin{array}{lr}

f(1)= &1\\
f(2)= &2\\
f(3)= &3\\
f(4)= &9\\
f(5)= &21\\
f(6)= &125\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Cancellative commutative partial monoids

Cancellative monoids

Partial monoids