### Table of Contents

## Boolean lattices

Abbreviation: **BoolLat**

### Definition

A \emph{Boolean lattice} is a bounded distributive lattice $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ such that

every element has a complement: $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be bounded distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ that is a bounded lattice homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$

### Examples

Example 1: $\langle \mathcal P(S), \cup, \emptyset, \cap, S\rangle$, the collection of subsets of a set $S$, with union, empty set, intersection, and the whole set $S$.

### Basic results

### Properties

Classtype | first-order |
---|---|

Equational theory | decidable |

Quasiequational theory | decidable |

First-order theory | decidable |

Congruence distributive | yes |

Congruence modular | yes |

Congruence n-permutable | yes |

Congruence regular | yes |

Congruence uniform | yes |

Congruence extension property | yes |

Definable principal congruences | yes |

Equationally def. pr. cong. | |

Amalgamation property | |

Strong amalgamation property | |

Epimorphisms are surjective | |

Locally finite | yes |

Residual size |

### Finite members

Any finite member is a power of the 2-element Boolean lattice.