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+% Introductory presentation:intropres.tex
+
+\documentclass{foils}
+\usepackage{amsmath}
+\usepackage{amssymb,latexsym}
+\usepackage{graphicx}
+
+\begin{document}
+\title{A construction of complete-simple\\
+ distributive lattices}
+\author{George~A. Menuhin\\
+ Computer Science Department\\
+ University of Winnebago\\
+ Winnebago, MN 53714}
+\date{March 15, 2006}
+
+\maketitle
+\begin{abstract}
+ In this presentation, we prove that there exist
+ \emph{complete-simple distributive lattices,}
+ that is, complete distributive lattices
+ with only two complete congruences.
+\end{abstract}
+
+\foilhead{The result}
+%\section{Introduction}\label{S:intro}
+In this presentation, 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}
+
+\foilhead{The construction}
+%\section{The $\Pi^{*}$ construction}\label{S:P*}
+The following construction is crucial in the proof
+of our theorem:
+
+\begin{Definition}\label{D:P*}
+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}
+
+\foilhead{}
+
+\begin{figure}[hbt]
+\centering\includegraphics[scale=2]{products}
+\caption{The product construction illustrated}\label{Fi:products}
+\end{figure}
+
+\foilhead{Notation}
+
+
+If $i \in I$ and $d \in D_{i}^{-}$, then
+\[
+ \langle \dots, 0, \dots, d, \dots, 0, \dots \rangle
+\]
+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.
+
+\foilhead{Second result}
+
+\begin{Theorem}\label{T:P*}
+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*}\label{E:cong1}
+ \langle \dots, d, \dots, 0, \dots \rangle \equiv
+ \langle \dots, c, \dots, 0, \dots \rangle
+ \pod{\Theta},
+\end{equation*}
+then $\Theta = \iota$.
+\end{Theorem}
+
+\foilhead{Verification}
+
+Since
+\begin{equation*}
+\langle \dots, d, \dots, 0, \dots \rangle \equiv
+\langle \dots, c, \dots, 0, \dots \rangle
+\pod{\Theta},
+\end{equation*}
+and $\Theta$ is a complete congruence relation,
+it follows from condition~(J) that modulo $\Theta$
+\begin{equation*}\label{E:cong}
+ \langle \dots, d, \dots, 0, \dots \rangle \equiv
+ \bigvee ( \langle \dots, c, \dots, 0, \dots \rangle
+ \mid d \leq c < 1 ).
+\end{equation*}
+
+\foilhead{Verification completed}
+
+Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
+Meeting both sides of the last congruence
+with $\langle \dots, a, \dots, 0, \dots \rangle$,
+we obtain that
+\begin{equation*}
+ 0 = \langle \dots, a, \dots, 0, \dots \rangle
+ \pod{\Theta},
+\end{equation*}
+Using the completeness of $\Theta$ and the previous equation,
+we get:
+\[
+ 0 \equiv \bigvee ( \langle \dots, a, \dots, 0,
+ \dots \rangle \mid a \in D_{j}^{-} ) = 1
+ \pod{\Theta},
+\]
+hence $\Theta = \iota$.
+
+\foilhead{}
+
+\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{document}
+