### Table of Contents

## Boolean groups

Abbreviation: **BGrp**

### Definition

A \emph{Boolean group} is a monoid $\mathbf{M}=\langle M, \cdot, e\rangle$ such that

every element has order $2$: $x\cdot x=e$.

##### Morphisms

Let $\mathbf{M}$ and $\mathbf{N}$ be Boolean groups. A morphism from $\mathbf{M}$ to $\mathbf{N}$ is a function $h:M\rightarrow N$ that is a homomorphism:

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

### Examples

Example 1: $\langle \{0,1\},+ ,0\rangle$, the two-element group with addition-mod-2. This algebra generates the variety of Boolean groups.

### Basic results

### Properties

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

Equational theory | decidable in polynomial time |

Quasiequational theory | decidable |

First-order theory | decidable |

Locally finite | yes |

Residual size | 2 |

Congruence distributive | no |

Congruence modular | yes |

Congruence n-permutable | yes, $n=2$ |

Congruence regular | yes |

Congruence uniform | yes |

Congruence extension property | yes |

Definable principal congruences | |

Equationally def. pr. cong. | no |

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &1

f(3)= &0

f(4)= &1

f(5)= &0

f(6)= &0

f(7)= &0

f(8)= &1

\end{array}$