Complemented lattices

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Changed: 5c5

A complemented lattice is a bounded lattice L = (L,∨,0,∧,1) such that

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variety

first-order

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decidable

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	Locally finite	no
Residual size	unbounded

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Locally finite	no
Residual size	unbounded

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[Size 5]?:

[Size 5]?: 2

Abbreviation: CdLat

Definition

A complemented lattice is a bounded 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 complemented 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) 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	first-order
Equational theory	decidable
Quasiequational theory
First-order theory	undecidable
Locally finite	no
Residual size	unbounded
Congruence distributive	yes
Congruence modular	yes
Congruence n-permutable	yes
Congruence regular	no
Congruence uniform	no
Congruence extension property	no
Definable principal congruences	no
Equationally definable principal congruences	no
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 0
[Size 4]?: 1
[Size 5]?: 2
[Size 6]?:
[Size 7]?:
[Size 8]?:

Subclasses

Complemented modular lattices

Superclasses

Bounded lattices