Mathematical Structures: History of Closure algebras

# History of Closure algebras

 %%run pdflatex % \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 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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{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} %
 %%run pdflatex % \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 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 $n$. $\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{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} %