History of Closure algebras

 
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 \begin{document}
 \textbf{\Large Closure algebras}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Closure_algebras}{edit}
 
 \abbreviation{CloA}
 
 \begin{definition}
 A \emph{closure algebra} is a modal algebra $\mathbf{A}=\langle A,\vee,0,
 \wedge,1,\neg,\diamond\rangle$ such that
 
 
 $\diamond$ is \emph{closure operator}:  
 $x\le \diamond x$, $\diamond\diamond x=\diamond x$
 
 Remark: 
 Closure algebras provide algebraic models for the modal logic S4. 
 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 closure 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}
 $\langle P(X),\cup,\emptyset,\cap,X,-,cl\rangle$, where $X$ is any topological space and $cl$ is the closure operator associated with $X$.
 \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 & yes\\\hline
 Equationally def. pr. cong. & yes\\\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 $n$.
 
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 \begin{subclasses}\ 
 
 \href{Monadic_algebras.pdf}{Monadic algebras} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Modal_algebras.pdf}{Modal algebras} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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