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\begin{document}
\title{A construction of complete-simple\\
distributive lattices}
\author{George~A. Menuhin\thanks{Research supported
by the NSF under grant number~23466.}\\
Computer Science Department\\
Winnebago, Minnesota 23714\\
[email protected]}
\date{March 15, 1999}
\maketitle
\begin{abstract}
In this note, we prove that there exist \emph{complete-simple
distributive lattices,} that is, complete distributive
lattices in which there are only two complete congruences.
\end{abstract}
\section{Introduction}\label{S:intro}
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}
\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}
\begin{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 $0$.
\end{notation}
See also Ernest~T. Moynahan~\cite{eM57a}.
Next we verify the following 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 \ldots, d, \ldots, 0, \ldots \rangle \equiv
\langle \ldots, c, \ldots, 0, \ldots \rangle \pmod{\Theta},
\end{equation}
then $\Theta = \iota$.
\end{theorem}
\emph{Proof.} Since
\begin{equation}\label{E:cong2}
\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}\label{E:cong}
\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}
Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
Meeting both sides of the congruence (\ref{E:cong2}) with
$\langle \ldots, a, \ldots, 0, \ldots \rangle$, we obtain that
\begin{equation}\label{E:comp}
0 = \langle \ldots, a, \ldots, 0, \ldots \rangle \pmod{\Theta},
\end{equation}
Using the completeness of $\Theta$ and (\ref{E:comp}),
we get:
\[
0 \equiv \bigvee ( \langle \ldots, a, \ldots, 0, \ldots \rangle
\mid a \in D_{j}^{-} ) = 1 \pmod{\Theta},
\]
hence $\Theta = \iota$.
\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-Toronto-London-Melbourne, 1968.
\bibitem{eM57}
Ernest~T. Moynahan,
\emph{On a problem of M.H. 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}
