Table of Contents
Tense algebras
Abbreviation: TA
Definition
A \emph{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 \emph{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 \emph{at some past instance} and \emph{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
$\begin{array}{lr}
f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$
