Algebraic lattices

 
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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 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}
 
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 \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}
 
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 \bibitem{Ln19xx}
 
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