Modal algebras

 
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 \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}
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 \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
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 \begin{document}
 \textbf{\Large Modal algebras}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Modal_algebras}{edit}
 
 \abbreviation{MA}
 
 \begin{definition}
 A \emph{modal algebra} is a structure $\mathbf{A}=\langle A,\vee,0,
 \wedge,1,\neg,\diamond\rangle$ such that
 
 
 $\langle A,\vee,0,
 \wedge,1,\neg\rangle $ is a \href{Boolean_algebras.pdf}{Boolean algebras}
 
 
 $\diamond$ is \emph{join-preserving}:  
 $\diamond(x\vee y)=\diamond x\vee \diamond y$
 
 
 $\diamond$ is \emph{normal}:  
 $\diamond 0=0$
 
 Remark: 
 Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the
 \emph{possibility operator}, and the \emph{necessity operator} $\Box$ is defined as $\Box x=\neg\diamond\neg x$.
 
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be modal algebras. 
 A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:
 
 $h(\diamond x)=\diamond h(x)$
 \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 $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &\\
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 \end{array}$
 \end{finite_members}
 \hyperbaseurl{http://math.chapman.edu/structures/files/}
 \parskip0pt
 \begin{subclasses}\ 
 
 \href{Closure_algebras.pdf}{Closure algebras} 
 
 \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}
 %

 
 \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 Modal algebras}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Modal_algebras}{edit}
 
 \abbreviation{MA}
 
 \begin{definition}
 A \emph{modal algebra} is a structure $\mathbf{A}=\langle A,\vee,0,
 \wedge,1,\neg,\diamond\rangle$ such that
 
 
 $\langle A,\vee,0,
 \wedge,1,\neg\rangle $ is a \href{Boolean_algebras.pdf}{Boolean algebras}
 
 
 $\diamond$ is \emph{join-preserving}:  
 $\diamond(x\vee y)=\diamond x\vee \diamond y$
 
 
 $\diamond$ is \emph{normal}:  
 $\diamond 0=0$
 
 Remark: 
 Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the
 \emph{possibility operator}, and the \emph{necessity operator} $\Box$ is defined as $\Box x=\neg\diamond\neg x$.
 
 \end{definition}
 
 \begin{morphisms}
 Let $\mathbf{A}$ and $\mathbf{B}$ be modal algebras. 
 A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:
 
 $h(\diamond x)=\diamond h(x)$
 \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{Closure_algebras.pdf}{Closure algebras} 
 
 \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/Modal_algebras.pdf

%%run pdflatex

%

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\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 Modal algebras}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Modal_algebras}{edit}

\abbreviation{MA}

\begin{definition}
A \emph{modal algebra} is a structure $\mathbf{A}=\langle A,\vee,0,
\wedge,1,\neg,\diamond\rangle$ such that


$\langle A,\vee,0,
\wedge,1,\neg\rangle $ is a \href{Boolean_algebras.pdf}{Boolean algebras}


$\diamond$ is \emph{join-preserving}:  
$\diamond(x\vee y)=\diamond x\vee \diamond y$


$\diamond$ is \emph{normal}:  
$\diamond 0=0$

Remark: 
Modal algebras provide algebraic models for modal logic. The operator $\diamond$ is the
\emph{possibility operator}, and the \emph{necessity operator} $\Box$ is defined as $\Box x=\neg\diamond\neg x$.

\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be modal algebras. 
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:

$h(\diamond x)=\diamond h(x)$
\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{Closure_algebras.pdf}{Closure algebras} 

\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}
%