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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, 1995}
\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 our proof of our Theorem:
\begin{definition} \label{D:P*}
Let $D_{i}$, $i \in I$, 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
\[
\langle \dots, 0, \dots, \overset{i}{d}, \dots, 0,
\dots \rangle
\]
is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose
$i$th component is $d$ and all the other components
are $0$.
\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 \dots, 0, \dots,\overset{i}{d},
\dots, 0, \dots \rangle \equiv \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta},
\end{equation}
then $\Theta = \iota$.
\end{theorem}
\emph{Proof.} Since
\begin{equation} \label{E:cong2}
\langle \dots, 0, \dots, \overset{i}{d}, \dots, 0,
\dots \rangle \equiv \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta},
\end{equation}
and $\Theta$ is a complete congruence relation, it follows
from condition~(C) that
\begin{align} \label{E:cong}
& \langle \dots, \overset{i}{d}, \dots, 0,
\dots \rangle \equiv\\
&\qquad \qquad \quad \bigvee ( \langle \dots, 0, \dots,
\overset{i}{c}, \dots, 0, \dots \rangle \mid d \leq c < 1 )
\equiv 1 \pmod{\Theta}. \notag
\end{align}
Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
Meeting both sides of the congruence \eqref{E:cong2} with
$\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0,
\dots \rangle$, we obtain
\begin{align} \label{E:comp}
0 = & \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, \dots
\rangle \wedge \langle \dots, 0, \dots, \overset{j}{a},
\dots, 0, \dots \rangle \equiv\\
&\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, \dots
\rangle \pmod{\Theta}, \notag
\end{align}
Using the completeness of $\Theta$ and \eqref{E:comp},
we get:
\[
0 \equiv \bigvee ( \langle \dots, 0, \dots, \overset{j}{a},
\dots, 0, \dots \rangle \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta},
\]
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-Mel\-bourne, 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}
