[Home]Modal algebras

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Abbreviation: MA


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


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



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]?:  


Closure algebras


Boolean algebras with operators

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Last edited March 25, 2003 9:46 pm (diff)