### Table of Contents

## Modal algebras

Abbreviation: **MA**

### Definition

A \emph{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 \emph{join-preserving}: $\diamond(x\vee y)=\diamond x\vee \diamond y$

$\diamond$ is \emph{normal}: $\diamond 0=0$

Remark: Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the \emph{possibility operator}, and the \emph{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}$