A modal algebra is a structure A = (A,∨,0,
∧,1,¬,◊) such that
(A,∨,0, ∧,1,¬) is a Boolean algebra,
◊ is join-preserving: ◊(x∨y) = ◊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).
|Congruence n-permutable||yes, n = 2|
|Congruence extension property||yes|
|Definable principal congruences||no|
|Equationally definable principal congruences||no|
|Strong amalgamation property||yes|
|Epimorphisms are surjective||yes|