### Table of Contents

## Generalized Boolean algebras

Abbreviation: **GBA**

### Definition

A \emph{generalized Boolean algebra} is a Brouwerian algebras $\mathbf{A}=\langle A, \vee, \wedge, 1, \rightarrow\rangle$ such that

$x\vee y=(x\rightarrow y)\rightarrow y$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be generalized Boolean algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)\ \mbox{and} h(1)=1$, $h(x\rightarrow y)=h(x)\rightarrow h(y)$

### Examples

Example 1:

### Basic results

### Properties

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

Equational theory | decidable |

Quasiequational theory | decidable |

First-order theory | decidable |

Locally finite | yes |

Residual size | $2$ |

Congruence distributive | yes |

Congruence modular | yes |

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

Congruence regular | yes |

Congruence e-regular | yes, $e=1$ |

Congruence uniform | yes |

Congruence extension property | yes |

Definable principal congruences | yes |

Equationally def. pr. cong. | yes |

Amalgamation property | yes |

Strong amalgamation property | yes |

Epimorphisms are surjective | yes |

### Finite members

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

f(2)= &1

f(3)= &0

f(4)= &1

f(5)= &0

f(6)= &0

\end{array}$