tense_algebras - MathStructures

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=====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====

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

====Subclasses====
[[]] 

====Superclasses====
[[Boolean algebras with operators]] 


====References====

[(Ln19xx>
)]

Trace: • boolean_semilattices • compact_topological_spaces • bands • m-zeroid • small_categories • groupoids • basic_logic_algebras • quantales • ordered_rings

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