Mathematical Structures: Complete lattices

# Complete lattices

http://mathcs.chapman.edu/structuresold/files/Complete_lattices.pdf
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\begin{document}
\textbf{\Large Complete lattices}

\abbreviation{CLat}
\begin{definition}
A \emph{complete lattice} is a structure $\mathbf{L}=\langle L,\bigvee,\bigwedge\rangle$ such that $\bigvee,\bigwedge$ map
subsets of $L$ to elements of $L$ and

$\langle L,\vee,\wedge\rangle$ is a \href{Lattices.pdf}{Lattices}

$\bigvee S$ is the least upper bound of $S$

$\bigwedge S$ is the greatest lower bound of $S$
\end{definition}
\begin{morphisms}
Let $\mathbf{L}$ and $\mathbf{M}$ be complete lattices.
A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ 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}
$\langle \mathcal{P}(X),\bigcup,\bigcap\rangle$, the set of all subsets of a set $X$, with union and intersection of families of sets.
\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_lattices.pdf}{Algebraic lattices}

\end{subclasses}
\begin{superclasses}\

\href{Lattices.pdf}{Lattices}

\end{superclasses}

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\bibitem{Ln19xx}

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\end{document}
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