Mathematical Structures: Tense algebras

# Tense algebras

http://mathcs.chapman.edu/structuresold/files/Tense_algebras.pdf
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\begin{document}
\textbf{\Large Tense algebras}

\abbreviation{TA}

\begin{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 \href{Modal_algebras.pdf}{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}.

\end{definition}

\begin{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 $\diamond_f$:

$h(\diamond x)=\diamond h(x)$
\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
\end{example}
\end{examples}

\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & variety\\\hline
Equational theory & decidable\\\hline
Quasiequational theory & decidable\\\hline
First-order theory & undecidable\\\hline
Locally finite & no\\\hline
Residual size & unbounded\\\hline
Congruence distributive & yes\\\hline
Congruence modular & yes\\\hline
Congruence n-permutable & yes, $n=2$\\\hline
Congruence regular & yes\\\hline
Congruence uniform & yes\\\hline
Congruence extension property & yes\\\hline
Definable principal congruences & no\\\hline
Equationally def. pr. cong. & no\\\hline
Discriminator variety & no\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

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

\href{.pdf}{}

\end{subclasses}
\begin{superclasses}\

\href{Boolean_algebras_with_operators.pdf}{Boolean algebras with operators}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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