Closure algebras

Abbreviation: CloA

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).

Some results

Examples

(P(X),∪,Ø,∩,X,−,cl), where X is any topological space and cl is the closure operator associated with X.

Properties

Finite members

Subclasses

Superclasses