Mathematical Structures: Algebraic lattices

# Algebraic lattices

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 %%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 Algebraic Lattices} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Algebraic_Lattices}{edit} \abbreviation{ALat} \begin{definition} An \emph{algebraic lattice} is a \href{Complete_lattices.pdf}{complete lattices} $\mathbf{A}=\langle A,\bigvee,\bigwedge\rangle$ such that every element is a join of compact elements An element $c\in A$ is \emph{compact} if for every subset $S\subseteq A$ such that $c\le\bigvee S$, there exists a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be algebraic lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a complete homomorphism: $h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$ \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 & second-order\\\hline Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline Epimorphisms are surjective & yes\\\hline \end{tabular} \end{properties} \end{table} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{Algebraic_distributive_lattices.pdf}{Algebraic distributive lattices} \end{subclasses} \begin{superclasses}\ \href{Complete_lattices.pdf}{Complete lattices} \href{Algebraic_semilattices.pdf}{Algebraic semilattices} \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 Algebraic Lattices} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Algebraic_Lattices}{edit} \abbreviation{ALat} \begin{definition} An \emph{algebraic lattice} is a \href{Complete_lattices.pdf}{complete lattice} $\mathbf{A}=\langle A,\bigvee,\bigwedge\rangle$ such that every element is a join of compact elements. An element $c\in A$ is \emph{compact} if for every subset $S\subseteq A$ such that $c\le\bigvee S$, there exists a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be algebraic lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a complete homomorphism: $h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$ \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 & second-order\\\hline Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline Epimorphisms are surjective & yes\\\hline \end{tabular} \end{properties} \end{table} \hyperbaseurl{http://math.chapman.edu/structures/files/} \begin{subclasses}\ \href{Algebraic_distributive_lattices.pdf}{Algebraic distributive lattices} \end{subclasses} \begin{superclasses}\ \href{Complete_lattices.pdf}{Complete lattices} \href{Algebraic_semilattices.pdf}{Algebraic semilattices} \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

http://mathcs.chapman.edu/structuresold/files/Algebraic_lattices.pdf
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\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\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}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Algebraic Lattices}

\abbreviation{ALat}

\begin{definition}
An \emph{algebraic lattice} is a \href{Complete_lattices.pdf}{complete lattice} $\mathbf{A}=\langle A,\bigvee,\bigwedge\rangle$ such that
every element is a join of compact elements.

An element $c\in A$ is \emph{compact} if for every subset $S\subseteq A$ such that $c\le\bigvee S$, there exists
a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$.
\end{definition}

\begin{morphisms}
Let $\mathbf{A}$ and $\mathbf{B}$ be algebraic lattices.
A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a complete homomorphism:

$h(\bigvee S)=\bigvee h[S] \mbox{ and } h(\bigwedge S)=\bigwedge h[S]$
\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 & second-order\\\hline
Amalgamation property & yes\\\hline
Strong amalgamation property & yes\\\hline
Epimorphisms are surjective & yes\\\hline
\end{tabular}
\end{properties}
\end{table}

\hyperbaseurl{http://math.chapman.edu/structures/files/}
\begin{subclasses}\

\href{Algebraic_distributive_lattices.pdf}{Algebraic distributive lattices}

\end{subclasses}

\begin{superclasses}\

\href{Complete_lattices.pdf}{Complete lattices}

\href{Algebraic_semilattices.pdf}{Algebraic semilattices}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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