Modal algebras
Abbreviation: MA
Definition
A modal algebra is a structure $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\diamond\rangle$ such that
$\langle A,\vee,0, \wedge,1,\neg\rangle $ is a Boolean algebras
$\diamond$ is join-preserving: $\diamond(x\vee y)=\diamond x\vee \diamond y$
$\diamond$ is normal: $\diamond 0=0$
Remark: Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the possibility operator, and the necessity operator $\Box$ is defined as $\Box x=\neg\diamond\neg x$.
Morphisms
Let $\mathbf{A}$ and $\mathbf{B}$ be modal algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:
$h(\diamond x)=\diamond h(x)$
Examples
Example 1:
Basic results
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 def. pr. cong. | no |
Discriminator variety | no |
Amalgamation property | yes |
Strong amalgamation property | yes |
Epimorphisms are surjective | yes |
Finite members
$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
Subclasses
Superclasses
References
Trace: » modal_algebras