Mathematical Structures: Tense algebras

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\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
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\newtheorem*{properties}{Properties}
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\begin{document}
\textbf{\Large Tense algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Tense_algebras}{edit}

\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)= &\\
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\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\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}
%

http://mathcs.chapman.edu/structuresold/files/Tense_algebras.pdf

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\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\addtolength{\oddsidemargin}{-.5in}
\addtolength{\evensidemargin}{-.5in}
\addtolength{\textwidth}{1in}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\pagestyle{myheadings}\thispagestyle{myheadings}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Tense algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Tense_algebras}{edit}

\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}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\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}
%