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## Groupoids

Abbreviation: BinOp

### Definition

A groupoid is a structure $\mathbf{A}=\langle A,\cdot\rangle$ where $\cdot$ is any binary operation on $A$.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be groupoids. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

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

Example 1:

### Properties

Classtype variety decidable undecidable no unbounded no no no no no no no no yes yes yes

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &10\\ f(3)= &3330\\ f(4)= &178981952\\ f(5)= &2483527537094825\\ f(6)= &14325590003318891522275680\\ f(7)= &50976900301814584087291487087214170039\\ f(8)= &155682086691137947272042502251643461917498835481022016\\ Michael A. Harrison,\emph{The number of isomorphism types of finite algebras}, Proc. Amer. Math. Soc., \textbf{17}1966,731--737[[http://www.ams.org/mathscinet-getitem?mr=34 :118|MRreview]] \end{array}$

### Subclasses

[[Commutative groupoids]]
[[Idempotent groupoids]]
[[Semigroups]]
[[Left-distributive groupoids]]