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\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem*{main}{Main~Theorem}
\newtheorem{lemma}{Lemma}
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\begin{document}
\title[Complete-simple distributive lattices]
     {A construction of complete-simple\\
      distributive lattices}
\author{George~A. Menuhin}
\address{Computer Science Department\\
        University of Winnebago\\
        Winnebago, Minnesota 53714}
\email{[email protected]}
\urladdr{http://math.uwinnebago.ca/homepages/menuhin/}
\thanks{Research supported by the NSF under grant number
~23466.}
\keywords{Complete lattice, distributive lattice,
complete congruence,
    congruence lattice}
\subjclass{Primary: 06B10; Secondary: 06D05}
\date{March 15, 1999}
\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}

\maketitle

\section{Introduction}\label{S:intro}
In this note we prove the following result:

\begin{main}
  There exists an infinite complete distributive lattice $K$ with only
  the two trivial complete congruence relations.
\end{main}

\section{The $D^{\langle 2 \rangle}$ construction}\label{S:Ds}
For the basic notation in lattice theory and universal algebra, see Ferenc~R.
Richardson~\cite{fR82} and George~A. Menuhin~\cite{gM68}.  We start with some
definitions:

\begin{definition}\label{D:prime}
  Let $V$ be a complete lattice, and let $\mathfrak{p} = [u, v]$ be
  an interval of $V$.  Then $\mathfrak{p}$ is called
  \emph{complete-prime} if the following three conditions are satisfied:
  \begin{itemize}
     \item[(1)] $u$ is meet-irreducible but $u$ is \emph{not}
        completely meet-irreducible;
     \item[(2)] $v$ is join-irreducible but $v$ is \emph{not}
        completely join-irreducible;
     \item[(3)] $[u, v]$ is a complete-simple lattice.
  \end{itemize}
\end{definition}

Now we prove the following result:

\begin{lemma}\label{L:ds}
  Let $D$ be a complete distributive lattice satisfying
  conditions~\textup{(1)} and~\textup{(2)}.  Then
  $D^{\langle 2 \rangle}$ is a sublattice of $D^{2}$;
  hence $D^{\langle 2 \rangle}$ is a lattice, and
  $D^{\langle 2 \rangle}$ is a complete distributive
  lattice satisfying conditions~\textup{(1)} and \textup{(2)}.
\end{lemma}

\begin{proof}
  By conditions~(1) and (2), $D^{\langle 2 \rangle}$ is a sublattice
  of $D^{2}$.  Hence, $D^{\langle 2 \rangle}$ is a lattice.

  Since $D^{\langle 2 \rangle}$ is a sublattice of a distributive
  lattice, $D^{\langle 2 \rangle}$ is a distributive lattice.  Using
  the characterization of standard ideals in Ernest~T. Moynahan~\cite{eM57},
  $D^{\langle 2 \rangle}$ has a zero and a unit element,
  namely, $\langle 0, 0 \rangle$ and $\langle 1, 1 \rangle$.
  To show that $D^{\langle 2 \rangle}$ is complete, let
  $\varnothing \ne A \subseteq D^{\langle 2 \rangle}$, and let
  $a = \bigvee A$ in $D^{2}$.  If
  $a \in D^{\langle 2 \rangle}$, then
  $a = \bigvee A$ in $D^{\langle 2 \rangle}$; otherwise, $a$
  is of the form $\langle b, 1 \rangle$ for some
  $b \in D$ with $b < 1$.  Now $\bigvee A = \langle 1, 1\rangle$
  in $D^{2}$ and the dual argument shows that $\bigwedge A$ also
  exists in $D^{2}$.  Hence $D$ is complete. Conditions~(1) and
  (2) are obvious for $D^{\langle 2 \rangle}$.
\end{proof}

\begin{corollary}\label{C:prime}
  If $D$ is complete-prime, then so is $D^{\langle 2 \rangle}$.
\end{corollary}

The motivation for the following result comes from Soo-Key Foo~\cite{sF90}.

\begin{lemma}\label{L:ccr}
  Let $\Theta$ be a complete congruence relation of
  $D^{\langle 2 \rangle}$ such that
  \begin{equation}\label{E:rigid}
     \langle 1, d \rangle \equiv \langle 1, 1 \rangle \pmod{\Theta},
  \end{equation}
  for some $d \in D$ with $d < 1$. Then $\Theta = \iota$.
\end{lemma}

\begin{proof}
  Let $\Theta$ be a complete congruence relation of
  $D^{\langle 2 \rangle}$ satisfying \eqref{E:rigid}. Then $\Theta =
\iota$.
\end{proof}

\section{The $\Pi^{*}$ construction}\label{S:P*}
The following construction is crucial to our proof of the Main Theorem:

\begin{definition}\label{D:P*}
  Let $D_{i}$, for $i \in I$, be complete distributive lattices
  satisfying condition~\textup{(2)}.  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:

\begin{theorem}\label{T:P*}
  Let $D_{i}$, for $i \in I$, be complete distributive lattices
  satisfying condition~\textup{(2)}.  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}

\begin{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~(3) that
  \begin{align}\label{E:cong}
     & \langle \dots, \overset{i}{d}, \dots, 0,
      \dots \rangle \equiv\\
     &\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 $j \in I$ for $j \neq i$, 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 $\Theta = \iota$.
\end{proof}

\begin{theorem}\label{T:P*a}
  Let $D_{i}$ for $i \in I$ be complete distributive lattices
  satisfying conditions \textup{(2)} and \textup{(3)}.  Then
  $\Pi^{*} ( D_{i} \mid i \in I )$ also satisfies conditions \textup{(2)}
   and \textup{(3)}.
\end{theorem}

\begin{proof}
  Let $\Theta$ be a complete congruence on
  $\Pi^{*} ( D_{i} \mid i \in I )$. Let $i \in I$.  Define
  \[
     \widehat{D}_{i} = \{ \langle \dots, 0, \dots, \overset{i}{d},
     \dots, 0, \dots \rangle \mid d \in D_{i}^{-} \} \cup \{ 1 \}.
  \]
  Then $\widehat{D}_{i}$ is a complete sublattice of
  $\Pi^{*} ( D_{i} \mid i \in I )$, and $\widehat{D}_{i}$ is
  isomorphic to $D_{i}$.  Let $\Theta_{i}$ be the restriction of
  $\Theta$ to $\widehat{D}_{i}$.

  Since $D_{i}\) is complete-simple, so is $\widehat{D}_{i}$, and
  hence $\Theta_{i}$ is $\omega$ or $\iota$.  If
  $\Theta_{i} = \rho$ for all $i \in I$, then
  $\Theta = \omega$.  If there is an $i \in I$, such that
  $\Theta_{i} = \iota$, then $0 \equiv 1 \pmod{\Theta}$, hence
  $\Theta = \iota$.
\end{proof}

The Main Theorem follows easily from Theorems~\ref{T:P*} and \ref{T:P*a}.

\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}
     \bysame, \emph{Ideals and congruence relations in lattices.~II,}
    Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} (1957),
    417--434  (Hungarian).

  \bibitem{fR82}
     Ferenc~R. Richardson, \emph{General Lattice Theory,} Mir, Moscow,
     expanded and revised ed., 1982 (Russian).

\end{thebibliography}
\end{document}

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