Boolean lattices

Abbreviation: BoolLat

Definition

A Boolean lattice is a bounded distributive lattice L = (L,∨,0,∧,1) such that
every element has a complement: ∃y(x∨y = 1 and x∧y = 0).

Morphisms

Let L and M be bounded distributive lattices. A morphism from L to M is a function h : L→M that is a bounded lattice homomorphism: h(x∨y) = h(x)∨h(y) and h(x∧y) = h(x)∧h(y) and h(0) = 0 and h(1) = 1.

Some results

Examples

(P(S),∪,Ø,∩,S), the collection of subsets of a set S, with union, empty set, intersection, and the whole set S.

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 definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective
Locally finite	yes
Residual size

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 0
[Size 4]?: 1
[Size 5]?:
[Size 6]?:
[Size 7]?:
[Size 8]?:

Subclasses

Superclasses

Complemented modular lattices
Bounded distributive lattices