Bounded distributive lattices

Abbreviation: BDLat

Definition

A bounded distributive lattice is a structure L = (L,∨,0,∧,1) such that
(L,∨,∧) is a distributive lattice,
0 is the least element: 0 ≤ x, and
1 is the greatest element: x ≤ 1.

Morphisms

Let L and M be bounded 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) 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	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	no
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 [Marcel Erne, Jobst Heitzig, Jürgen Reinhold, On the number of distributive lattices, Electron. J. Combin. 9 (2002) Research Paper 24, 23 pp. (electronic) MRreview]

Subclasses

Boolean algebras
[Complete distributive lattices]?

Superclasses

Distributive lattices
[Bounded modular lattices]?