Boolean lattices

Abbreviation: BoolLat

Definition

A 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 $S$.

Properties

Classtype first-order decidable decidable decidable yes yes yes yes yes yes yes yes

Finite members

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