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