Modal algebras

Abbreviation: MA

Definition

A modal algebra is a structure A = (A,∨,0, ∧,1,¬,◊) such that
(A,∨,0, ∧,1,¬) is a Boolean algebra,
◊ is join-preserving: ◊(x∨y) = ◊x∨◊y
◊ is normal: ◊0 = 0.

Remark: Modal algebras provide algebraic models for modal logic. The operator ◊ is the possibility operator, and the necessity operator □ is defined as □x = ¬◊¬x.

Morphisms

Let A and B be modal algebras. A morphism from A to B is a function h : A → B that is a Boolean homomorphism and preserves ◊: h(◊x) = ◊h(x).

Some results

Examples

Properties

Classtype	variety
Equational theory	decidable
Quasiequational theory	decidable
First-order theory	undecidable
Locally finite	no
Residual size	unbounded
Congruence distributive	yes
Congruence modular	yes
Congruence n-permutable	yes, n = 2
Congruence regular	yes
Congruence uniform	yes
Congruence extension property	yes
Definable principal congruences	no
Equationally definable principal congruences	no
[Discriminator variety]?	no
Amalgamation property	yes
Strong amalgamation property	yes
Epimorphisms are surjective	yes

Finite members

[Size 1]?: 1
[Size 2]?:
[Size 3]?:
[Size 4]?:
[Size 5]?:
[Size 6]?:

Subclasses

Closure algebras

Superclasses

Boolean algebras with operators