History of Complete semilattices

 
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 \begin{document}
 \textbf{\Large Complete semilattices}
 \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Complete_semilattices}{edit}
 
 \abbreviation{CSlat}
 \begin{definition}
 A \emph{complete semilattice} is a \href{Directed_complete_partial_orders.pdf}{directed complete partial orders} $\mathbf{P}=\langle P,\leq \rangle $
 such that every nonempty subset of $P$ has a greatest lower bound: 
 $\forall S\subseteq P\ (S\ne\emptyset\implies \exists z\in P(z=\bigwedge S))$.
 \end{definition}
 \begin{morphisms}
 Let $\mathbf{P}$ and $\mathbf{Q}$ be complete semilattices. A morphism from $\mathbf{P}$ to 
 $\mathbf{Q}$ is a function $f:P\rightarrow Q$ that preserves all nonempty meets and all directed joins: 
 
 $z=\bigwedge S\implies f(z)=\bigwedge f[S]$ for all nonempty $S\subseteq P$ and 
 $z=\bigvee D\implies f(z)= \bigvee f[D]$
 
 \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 $n$.
 
 $\begin{array}{lr}
 f(1)= &1\\
 f(2)= &\\
 f(3)= &\\
 f(4)= &\\
 f(5)= &\\
 f(6)= &\\
 \end{array}$
 \end{finite_members}
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 \begin{subclasses}\ 
 
 \href{Complete_lattices.pdf}{Complete lattices} 
 
 \end{subclasses}
 \begin{superclasses}\ 
 
 \href{Directed_complete_partial_orders.pdf}{Directed complete partial orders} 
 
 \end{superclasses}
 
 \begin{thebibliography}{10}
 
 \bibitem{Ln19xx}
 
 \end{thebibliography}
 
 \end{document}
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