Abbreviation: Pargoid

A \emph{partial groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$, where

$\cdot$ is a \emph{partial binary operation}, i.e., $\cdot: A\times A\to A+\{*\}$.

Remark: The domain of definition of $\cdot$ is Dom$(\cdot)=\{\langle x,y\rangle\in A^2 \mid x\cdot y\ne *\}$

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

Example 1: The empty partial binary operation on any set $A$ gives a partial groupoid.

$\begin{array}{lr}

f(1)= &2\\
f(2)= &45\\
f(3)= &43968\\
f(4)= &6358196250\\
f(5)= &236919104155855296\\

\end{array}$

See http://oeis.org/A090601

Groupoids

Partial semigroups

Ternary relations