### Table of Contents

## Commutative binars

Abbreviation: **CBin**

### Definition

A \emph{commutative binar} is a binar $\mathbf{A}=\langle A,\cdot\rangle$ such that

$\cdot$ is commutative: $x\cdot y=y\cdot x$.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be commutative binars. 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: $\langle\mathbb N,|\cdot|\rangle$ is the distance binar of the natural numbers, where the binary operation is $|x-y|$.

### 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

n | # of algebras |
---|---|

1 | 1 |

2 | 4 |

3 | 129 |

4 | 43968 |

5 | 254429900 |

6 | 30468670170912 |

7 | 91267244789189735259 |

8 | 8048575431238519331999571800 |

9 | 24051927835861852500932966021650993560 |

10 | 2755731922430783367615449408031031255131879354330 |

see finite commutative binars and http://www.research.att.com/~njas/sequences/A001425