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.
