Closure algebras

HomePage | RecentChanges | Preferences

Definition

A closure algebra is a modal algebra A = (A,∨,0, ∧,1,¬,◊) such that
is closure operator:   x ≤ ◊x  and  ◊◊x = ◊x.

Remark: Closure algebras provide algebraic models for the modal logic S4. The operator is the possibility operator, and the necessity operator is defined as x = ¬◊¬x.

Morphisms

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

Examples

(P(X),∪,Ø,∩,X,−,cl), where X is any topological space and cl is the closure operator associated with 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 yes Equationally definable principal congruences yes [Discriminator variety]? no Amalgamation property yes Strong amalgamation property yes Epimorphisms are surjective yes

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