# Modal algebras

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### Definition

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.

### 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).

### 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

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Closure algebras

### Superclasses

Boolean algebras with operators

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