Differences

This shows you the differences between two versions of the page.

tense_algebras [2010/07/29 15:46] (current)
Line 1: Line 1:
 +=====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>
 +)]
 +
 +
 +
 +
 +
 +