## 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 $\diamond_f$:

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

Example 1:

### Properties

Classtype variety decidable decidable undecidable no unbounded yes yes yes, $n=2$ yes yes yes no no no yes yes yes

### Finite members

$\begin{array}{lr} f(1)= &1 f(2)= & f(3)= & f(4)= & f(5)= & f(6)= & \end{array}$

### Superclasses

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