\begin{document}
\title{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}
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}
\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}
\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}
\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{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.
