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Table of Contents

## 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:

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

### Examples

Example 1:

### Basic results

### Properties

Classtype | variety |
---|---|

Equational theory | decidable |

Quasiequational theory | |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | no |

Congruence modular | no |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | no |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | yes |

Strong amalgamation property | yes |

Epimorphisms are surjective | yes |

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &10\\ f(3)= &3330\\ f(4)= &178981952\\ f(5)= &2483527537094825\\ f(6)= &14325590003318891522275680\\ f(7)= &50976900301814584087291487087214170039\\ f(8)= &155682086691137947272042502251643461917498835481022016 \end{array}$

### Subclasses

[[Commutative groupoids]]

[[Idempotent groupoids]]

[[Semigroups]]

[[Left-distributive groupoids]]

### Superclasses

### References

Trace: » groupoids