Bounded lattices

Abbreviation: BLat

Definition

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

Morphisms

Let L and M be bounded 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	no
Definable principal congruences	no
Equationally definable principal congruences	no
Amalgamation property	yes
Strong amalgamation property	yes
Epimorphisms are surjective	yes
Locally finite	no
Residual size	unbounded

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 1
[Size 4]?: 2
[Size 5]?: 5
[Size 6]?: 15
[Size 7]?: 53
[Size 8]?: 222
[Size 9]?: 1078
[Size 10]?: 5994
[Size 11]?: 37622
[Size 12]?: 262776
[Size 13]?: 2018305
[Size 14]?: 16873364
[Size 15]?: 152233518
[Size 16]?: 1471613387
[Size 17]?: 15150569446
[Size 18]?: 165269824761

[Jobst Heitzig, Jürgen Reinhold, Counting finite lattices, Algebra Universalis 48 (2002) 43--53 MRreview]

Subclasses

[Bounded modular lattices]?
Complete lattices

Superclasses

Lattices