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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}
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