### Table of Contents

## Binars

Abbreviation: **Bin**

### Definition

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

Remark: In Universal Algebra binars are also called \emph{groupoids}. However the more common usage of this term now refers to a category in which each morphism is an isomorphism.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be 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: $\langle\mathbb N,{}^\wedge\rangle$ is the exponentiation binar of the natural numbers, where $0{}^\wedge0=1$. It is not associative nor commutative, and does not have a (two-sided) identity.

### 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 | 10 |

3 | 3330 |

4 | 178981952 |

5 | 2483527537094825 |

6 | 14325590003318891522275680 |

7 | 50976900301814584087291487087214170039 |

8 | 155682086691137947272042502251643461917498835481022016 |

Michael A. Harrison, \emph{The number of isomorphism types of finite algebras}, Proc. Amer. Math. Soc., \textbf{17} 1966, 731–737 MRreview