\begin{document}
\title{A construction of complete-simple\\
distributive lattices}
\author{George~A. Menuhin}
\address{Computer Science Department\\
University of Winnebago\\
Winnebago, MN 53714}
\date{March 15, 2006}
\begin{abstract}
In this note, we prove that there exist
\emph{complete-simple distributive lattices,}
that is, complete distributive lattices
with only two complete congruences.
\end{abstract}
\maketitle
\section{Introduction}\label{S:intro}
In this note we prove the following result:
\begin{theorem}\index{Main 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*}
\index{pistar@$\Pi^{*}$ construction}%
\index{Main Theorem!exposition|(}%
The following construction is crucial in the proof
of our Theorem (see Figure~\ref{Fi:products}):
\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
\[
\langle \dots, 0, \dots, d, \dots, 0, \dots \rangle
\]
is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose
$i$-th component is $d$ and all the other components
are $0$.
\end{notation}
See also Ernest~T.
\index{Moynahan, Ernest~T.}%
Moynahan~\cite{eM57a}.
Next we verify the following result:
\index{lattice|textbf}%
\index{lattice!distributive}%
\index{lattice!distributive!complete}%
\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}
\begin{proof}
Since
\begin{equation}\label{E:cong2}
\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
\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 ) \pod{\Theta}.
\end{equation}
Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
Meeting both sides of the congruence \eqref{E:cong2}
with $\langle \dots, a, \dots, 0, \dots \rangle$,
we obtain that
\begin{equation}\label{E:comp}
0 = \langle \dots, a, \dots, 0, \dots \rangle
\pod{\Theta},
\end{equation}
Using the completeness of $\Theta$ and \eqref{E:comp},
we get:
\[
0 \equiv \bigvee ( \langle \dots, a, \dots, 0,
\dots \rangle \mid a \in D_{j}^{-} ) = 1
\pod{\Theta},
\]
hence $\Theta = \iota$.
\index{Main Theorem!exposition|)}
\end{proof}
\begin{thebibliography}{9}
\bibitem{sF90}\index{Foo, Soo-Key}%
Soo-Key Foo,
\emph{Lattice Constructions},
Ph.D. thesis,
University of Winnebago, Winnebago, MN, December, 1990.
\bibitem{eM57}\index{Moynahan, Ernest~T.}%
Ernest~T. Moynahan,
\emph{On a problem of M. Stone},
Acta Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460.
\bibitem{eM57a}\index{Moynahan, Ernest~T.}%
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}
