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.
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).
(P(X),∪,Ø,∩,X,−,cl), where X is any topological space and cl is the closure operator associated with X.
|Congruence n-permutable||yes, n = 2|
|Congruence extension property||yes|
|Definable principal congruences||yes|
|Equationally definable principal congruences||yes|
|Strong amalgamation property||yes|
|Epimorphisms are surjective||yes|