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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: $\exists D\subseteq A\times A(\cdot:D\to A)$.

Remark: $x\cdot y\in A\iff \langle x,y\rangle\in D$

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 $x\cdot y\in A$ 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

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

Classtype first-order

Finite members

$\begin{array}{lr} f(1)= &2\\ f(2)= &45\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ \end{array}$ $\begin{array}{lr} f(6)= &\\ f(7)= &\\ f(8)= &\\ f(9)= &\\ f(10)= &\\ \end{array}$

Subclasses

[[Groupoids]]
[[Partial semigroups]]

Superclasses

[[Ternary relations]]