## Tense algebras

Abbreviation: **TA**

### Definition

A ** tense algebra** is a structure $\mathbf{A}=\langle A,\vee,0,
\wedge,1,\neg,\diamond_f, \diamond_p\rangle$ such that both

$\langle A,\vee,0,\wedge,1,\neg,\diamond_f\rangle$ and $\langle A,\vee,0,\wedge,1,\neg,\diamond_p\rangle$ are Modal algebras

$\diamond_p$ and $\diamond_f$ are ** conjugates**:
$x\wedge\diamond_py = 0$ iff $\diamond_fx\wedge y = 0$

Remark:
Tense algebras provide algebraic models for logic of tenses. The two possibility operators
$\diamond_p$ and $\diamond_f$ are intuitively interpreted as
** at some past instance** and

**.**

*at some future instance*##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be tense algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond_p$ and $\diamond_f$:

$h(\diamond x)=\diamond h(x)$

### 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

### References

Trace: » tense_algebras