=====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 $x\cdot e=x=e\cdot x$.

====Basic results====


====Properties====

^[[Classtype]]                        |first-order  |
^[[Equational theory]]                | |
^[[Quasiequational theory]]           | |
^[[First-order theory]]               | |
^[[Locally finite]]                   | |
^[[Residual size]]                    | |
^[[Congruence distributive]]          | |
^[[Congruence modular]]               | |
^[[Congruence $n$-permutable]]        | |
^[[Congruence regular]]               | |
^[[Congruence uniform]]               | |
^[[Congruence extension property]]    | |
^[[Definable principal congruences]]  | |
^[[Equationally def. pr. cong.]]      | |
^[[Amalgamation property]]            | |
^[[Strong amalgamation property]]     | |
^[[Epimorphisms are surjective]]      | |

====Finite members====

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

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


====Subclasses====
[[Partial commutative monoids]]

[[Monoids]]


====Superclasses====
[[Partial semigroups]]


====References====