# Distributive lattices

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### Definition

A distributive lattice is a lattice L = (L,∨,∧) such that
distributes over :   x∧( yz)  = ( xy) ∨( xz) .

### Definition

A distributive lattice is a lattice L = (L,∨,∧) such that
distributes over :   x∨( yz)  = ( xy) ∧( xz) .

### Definition

A distributive lattice is a lattice L = (L,∨,∧) such that
( xy) ∨( xz) ∨( yz)  = ( xy) ∧( xz) ∧( yz) .

### 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 : LM that is a homomorphism: h(xy) = h(x)∨h(y)  and  h(xy) = h(x)∧h(y).

### Examples

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

### Properties

 Classtype variety 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 yes, \$\langle c,d\rangle\in Cg(a,b)\iff (a\wedge b)\wedge c=(a\wedge b)\wedge d \mbox{and} (a\vee b)\vee c=(a\vee b)\vee d Amalgamation property yes Strong amalgamation property no Epimorphisms 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

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