## Partial monoids

Abbreviation: PMon

### Definition

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 $x\in A$.

##### Morphisms

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 $h(x \cdot y)=h(x) \cdot h(y)$.

### Examples

Example 1: Any partial semigroup with a new element $e$ and $\cdot$ extended with $x\cdot e=x=e\cdot x$.

### Properties

Classtype first-order

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &3\\ f(3)= &15\\ f(4)= &112\\ f(5)= &\\ f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$