Distributive lattices

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Abbreviation: DLat

Definition

A distributive lattice is a lattice L = (L,∨,∧) such that
∧ distributes over ∨: x∧( y∨z) = ( x∧y) ∨( x∧z) .

Definition

A distributive lattice is a lattice L = (L,∨,∧) such that
∨ distributes over ∧: x∨( y∧z) = ( x∨y) ∧( x∨z) .

Definition

A distributive lattice is a lattice L = (L,∨,∧) such that
( x∧y) ∨( x∧z) ∨( y∧z) = ( x∨y) ∧( x∨z) ∧( y∨z) .

Definition

A distributive lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic to the diamond M₃ or the pentagon N₅

Morphisms

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

Some results

Examples

(P(S),∪,∩, ⊆ ), the collection of subsets of a sets S, ordered by inclusion.

Properties

Equational theory	Decidable
Quasiequational theory	Decidable
First-order theory	Undecidable
Congruence distributive	Yes
Congruence modular	Yes
Congruence n-permutable	No
Congruence regular	No
Congruence uniform	No
Congruence extension property	Yes
Definable principal congruences	No
Equationally definable principal congruences	No
Amalgamation property	Yes
Strong amalgamation property	No
[Epimorhisms are surjective]?	No
Locally finite	Yes
Residual size	2

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 1
[Size 4]?: 2
[Size 5]?: 3
[Size 6]?: 5
[Size 7]?: 8
[Size 8]?: 15
[Size 9]?: 26
[Size 10]?: 47
[Size 11]?: 82
[Size 12]?: 151
[Size 13]?: 269
[Size 14]?: 494
[Size 15]?: 891
[Size 16]?: 1639
[Size 17]?: 2978
[Size 18]?: 5483
[Size 19]?: 10006
[Size 20]?: 18428
Values known up to size 49 [Erne, Heitzig, Reinhold (2002)]

Subclasses

[One-element algebras]?
Bounded distributive lattices
[Complete distributive lattices]?

Superclasses

Modular lattices
Semidistributive lattices