History of Algebraic semilattices

 
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 \begin{document}
 \textbf{\Large Algebraic semilattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Algebraic_semilattices}{edit}
 
 \abbreviation{ASlat}
 \begin{definition}
 An \emph{algebraic semilattice} is a \href{Complete_semilattices.pdf}{complete semilattices} $\mathbf{P}=\langle P,\leq \rangle $
 such that
 
 the set of compact elements below any element is directed and
 
 every element is the join of all compact elements below it.
 
 An element $c\in P$ is \emph{compact} if for every subset $S\subseteq P$ such that $c\le\bigvee S$, there exists
 a finite subset $S_0$ of $S$ such that $c\le\bigvee S_0$.
 
 The set of compact elements of $P$ is denoted by $K(P)$.
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{P}$ and $\mathbf{Q}$ be algebraic semilattices. A morphism from $\mathbf{P}$ to 
 $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is \emph{Scott-continuous}, which means that $f$ preserves all directed joins: 
 
 $z=\bigvee D\implies f(z)= \bigvee f[D]$
 
 \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 & \\\hline
 Strong amalgamation property & \\\hline
 Epimorphisms are surjective & \\\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{Algebraic_lattices.pdf}{Algebraic lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Algebraic_posets.pdf}{Algebraic posets} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
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