### Table of Contents

## Commutative Groupoids

Abbreviation: **CBinOp**

### Definition

A \emph{commutative groupoid} is a structure $\mathbf{A}=\langle A,\cdot\rangle$ where $\cdot$ is any commutative binary operation on $A$, i.e. $x\cdot y=y\cdot x$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be commutative 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)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\

\end{array}$

### Subclasses

[[Commutative semigroups]]

[[Idempotent commutative groupoids]]

[[Commutative left-distributive groupoids]]

### Superclasses

[[Groupoids]]