Algebraic lattices

http://math.chapman.edu/structuresold/files/Algebraic_lattices.pdf

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\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}

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\end{document}
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