−Table of Contents
Boolean lattices
Abbreviation: BoolLat
Definition
A \emph{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), h(x∧y)=h(x)∧h(y), h(0)=0, h(1)=1
Examples
Example 1: ⟨P(S),∪,∅,∩,S⟩, 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.