Abbreviation: PMon

A \emph{partial monoid} is a structure $\mathbf{A}=\langle A,\cdot,e\rangle$, where $\langle A,\cdot\rangle$ is a partial semigroup and

$e$ is an identity for $\cdot$: $x\cdot e=x=e\cdot x$ for all $x\in A$.

Let $\mathbf{A}$ and $\mathbf{B}$ be 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: Any partial semigroup with a new element $e$ and $\cdot$ extended with $x\cdot e=x=e\cdot x$.

http://mathv.chapman.edu/~jipsen/uajs/PMon.html

$\begin{array}{lr}

f(1)= &1\\
f(2)= &3\\
f(3)= &15\\
f(4)= &112\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
f(8)= &\\
f(9)= &\\
f(10)= &\\

\end{array}$

Partial commutative monoids

Monoids

Partial semigroups