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+%Introductory beamer presentation: quickbeamer2.tex
+
+\documentclass{beamer}
+\usetheme{Berkeley}
+
+\begin{document}
+\title[Complete-simple distributive lattices]{A construction of complete-simple\\ distributive lattices}
+\author[]{George~A. Menuhin}
+\institute{Computer Science Department\\
+ University of Winnebago\\
+ Winnebago, MN 53714}
+\date{March 15, 2006}
+
+\begin{frame}
+\titlepage
+\end{frame}
+
+\begin{frame}
+\frametitle{Outline}
+
+\tableofcontents[pausesections]
+\end{frame}
+
+\section{Introduction}
+
+\begin{frame}
+\frametitle{Introduction}
+
+In this note, we prove the following result:
+
+\begin{theorem}
+There exists an infinite complete distributive
+lattice~$K$ with only the two trivial complete
+congruence relations.
+\end{theorem}
+\end{frame}
+
+\section[Construction]{The $\Pi^{*}$ construction}
+
+\begin{frame}
+\frametitle{The $\Pi^{*}$ construction}
+
+The following construction is crucial in the proof
+of our Theorem:
+
+\begin{definition}
+Let $D_{i}$, for $i \in I$, be complete distributive
+lattices satisfying condition~\textup{(J)}. Their
+$\Pi^{*}$ product is defined as follows:
+\[
+ \Pi^{*} ( D_{i} \mid i \in I ) =
+ \Pi ( D_{i}^{-} \mid i \in I ) + 1;
+\]
+that is, $\Pi^{*} ( D_{i} \mid i \in I )$ is
+$\Pi ( D_{i}^{-} \mid i \in I )$ with a new
+unit element.
+\end{definition}
+\end{frame}
+
+\begin{frame}
+\frametitle{Illustrating the construction}
+
+\centering\includegraphics{products}
+\end{frame}
+
+\begin{frame}
+\frametitle{Notation}
+
+If $i \in I$ and $d \in D_{i}^{-}$, then
+\[
+ \langle \ldots, 0, \ldots, d, \ldots, 0, \ldots \rangle
+\]
+is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose
+$i$-th component is $d$ and all the other components
+are $0$.
+
+See also Ernest~T. Moynahan, 1957.
+\end{frame}
+
+\section[Second result]{The second result}
+
+\begin{frame}
+\frametitle{The second result}
+
+Next we verify the following result:
+
+\begin{theorem}
+Let $D_{i}$, $i \in I$, be complete distributive
+lattices satisfying condition~\textup{(J)}.
+Let $\Theta$ be a complete congruence relation on
+$\Pi^{*} ( D_{i} \mid i \in I )$.
+If there exist $i \in I$ and $d \in D_{i}$ with
+$d < 1_{i}$ such that, for all $d \leq c < 1_{i}$,
+\begin{equation*}
+ \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv
+ \langle \ldots, c, \ldots, 0, \ldots \rangle
+ \pmod{\Theta},
+\end{equation*}
+then $\Theta = \iota$.
+\end{theorem}
+\end{frame}
+
+\section{Proof}
+
+\begin{frame}
+\frametitle{Starting the proof}
+
+Since
+\begin{equation*}
+\langle \ldots, d, \ldots, 0, \ldots \rangle \equiv
+\langle \ldots, c, \ldots, 0, \ldots \rangle
+\pmod{\Theta},
+\end{equation*}
+and $\Theta$ is a complete congruence relation,
+it follows from condition~(J) that
+\begin{equation*}
+ \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv
+ \bigvee ( \langle \ldots, c, \ldots, 0, \ldots \rangle
+ \mid d \leq c < 1 ) \pmod{\Theta}.
+\end{equation*}
+\end{frame}
+
+\begin{frame}
+\frametitle{Completing the proof}
+
+Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
+Meeting both sides of the congruence
+with $\langle \ldots, a, \ldots, 0, \ldots \rangle$,
+we obtain that
+\begin{equation*}
+ 0 = \langle \ldots, a, \ldots, 0, \ldots \rangle
+ \pmod{\Theta},
+\end{equation*}
+Using the completeness of $\Theta$ and the penultimate equation,
+we get:
+\[
+ 0 \equiv \bigvee ( \langle \ldots, a, \ldots, 0,
+ \ldots \rangle \mid a \in D_{j}^{-} ) = 1
+ \pmod{\Theta},
+\]
+hence $\Theta = \iota$.
+\end{frame}
+
+\section{References}
+
+\begin{frame}
+\frametitle{References}
+
+\begin{thebibliography}{9}
+
+\bibitem{sF90}
+Soo-Key Foo,
+\emph{Lattice Constructions},
+Ph.D. thesis,
+University of Winnebago, Winnebago, MN, December, 1990.
+
+\bibitem{gM68}
+George~A. Menuhin,
+\emph{Universal Algebra},
+D.~van Nostrand, Princeton, 1968.
+
+\bibitem{eM57}
+Ernest~T. Moynahan,
+\emph{On a problem of M. Stone},
+Acta Math. Acad. Sci. Hungar. \textbf{8} (1957),
+455--460.
+
+\bibitem{eM57a}
+Ernest~T. Moynahan,
+\emph{Ideals and congruence relations in lattices.} II,
+Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9}
+(1957), 417--434.
+
+\end{thebibliography}
+\end{frame}
+\end{document}
+