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 $\diamond_f$:

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

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

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