Partial groupoids

Abbreviation: Pargoid

Definition

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$(\cdot)=\{\langle x,y\rangle\in A^2 \mid x\cdot y\ne *\}$

Morphisms

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

Examples

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

Properties

Classtype first-order

Finite members

$\begin{array}{lr} f(1)= &2\\ f(2)= &45\\ f(3)= &43968\\ f(4)= &6358196250\\ f(5)= &236919104155855296\\ \end{array}$