De Morgan algebras

Abbreviation: DeMA

Definition

A De Morgan algebra is a structure A = (A,∨,0,∧,1,¬) such that
(A,∨,0,∧,1) is a bounded distributive lattice, and
¬ is a De Morgan involution: ¬( x∧y) = ¬x∨¬y and ¬¬x = x.

Remark: It follows that ¬( x∨y) = ¬x∧¬y, ¬1 = 0 and ¬0 = 1 (e.g. ¬1 = ¬1∨0 = ¬1∨¬¬0 = ¬(1∧¬0) = ¬¬0 = 0). Thus ¬ is a dual automorphism.

Morphisms

Let A and B be De Morgan algebras. A morphism from A to B is a function h : A→B that is a homomorphism: h(x∨y) = h(x)∨h(y) and h(¬x) = ¬h(x).

Some results

Examples

Properties

Classtype	Variety
Equational theory
Quasiequational theory
First-order theory
Congruence distributive	Yes
Congruence modular	Yes
Congruence n-permutable
Congruence regular
Congruence uniform
Congruence extension property	Yes
Definable principal congruences
Equationally definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective
Locally finite
Residual size

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 2
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Subclasses

Boolean algebras

Superclasses

Ockham algebras