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Partial groupoids

Abbreviation: Pargoid

Definition

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

$\cdot$ is a 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

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