% Sample file: sampart2.tex
% The sample article for the amsart document class
% with user-defined commands
% Typeset with LaTeX format
\documentclass{amsart}
\usepackage{amssymb,latexsym}
\usepackage{lattice}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}{Corollary}
\newtheorem*{main}{Main~Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\theoremstyle{remark}
\newtheorem*{notation}{Notation}
\numberwithin{equation}{section}
\newcommand{\Prodm}[2]{\gP(\,#1\mid#2\,)}
% product with a middle
\newcommand{\Prodsm}[2]{\gP^{*}(\,#1\mid#2\,)}
% product * with a middle
\newcommand{\vct}[2]{\vv<\dots,0,\dots,\overset{#1}{#2},%
\dots,0,\dots>}% special vector
\newcommand{\fp}{\F{p}}% Fraktur p
\newcommand{\Ds}{D^{\langle2\rangle}}
\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 23714}
\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, 1995}
\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 $\Ds$ 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 $\fp = [u, v]$ be
an interval of $V$. Then $\fp$ is called
\emph{complete-prime} if the following three conditions are satisfied:
\begin{enumerate}
\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{enumerate}
\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 $\Ds$ is a sublattice of $D^{2}$; hence $\Ds$ is
a lattice, and $\Ds$ is a complete distributive lattice
satisfying conditions~~\textup{(1)} and~~\textup{(2)}.
\end{lemma}
\begin{proof}
By conditions~(1) and (2), $\Ds$ is a sublattice of
$D^{2}$. Hence, $\Ds$ is a lattice.
Since $\Ds$ is a sublattice of a distributive lattice, $\Ds$ is
a distributive lattice. Using the characterization of
standard ideals in Ernest~T. Moynahan~\cite{eM57},
$\Ds$ has a zero and a unit element, namely,
$\vv<0, 0>$ and $\vv<1, 1>$. To show that $\Ds$ is
complete, let $\es \ne A \ci \Ds$, and let $a = \JJ A$
in $D^{2}$. If $a \in \Ds$, then
$a = \JJ A$ in $\Ds$; otherwise, $a$ is of the form
$\vv<b, 1>$ for some $b \in D$ with $b < 1$. Now
$\JJ A = \vv<1, 1>$ in $D^{2}$, and
the dual argument shows that $\MM A$ also exists in
$D^{2}$. Hence $D$ is complete. Conditions~(1) and (2)
are obvious for $\Ds$.
\end{proof}
\begin{corollary}\label{C:prime}
If $D$ is complete-prime, then so is $\Ds$.
\end{corollary}
The motivation for the following result comes from Soo-Key
Foo~\cite{sF90}.
\begin{lemma}\label{L:ccr}
Let $\gQ$ be a complete congruence relation of $\Ds$ such
that
\begin{equation}\label{E:rigid}
\vv<1, d> \equiv \vv<1, 1> \pod{\gQ},
\end{equation}
for some $d \in D$ with $d < 1$. Then $\gQ = \gi$.
\end{lemma}
\begin{proof}
Let $\gQ$ be a complete congruence relation of $\Ds$
satisfying \eqref{E:rigid}. Then $\gQ = \gi$.
\end{proof}
\section{The $\gP^{*}$ 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~\tup{(2)}. Their $\gP^{*}$
product is defined as follows:
\[
\Prodsm{ D_{i} }{i \in I} = \Prodm{ D_{i}^{-} }{i \in I} +1;
\]
that is, $\Prodsm{ D_{i} }{i \in I}$ is
$\Prodm{ D_{i}^{-} }{i \in I}$ with a new unit element.
\end{definition}
\begin{notation}
If $i \in I$ and $d \in D_{i}^{-}$, then
\[
\vct{i}{d}
\]
is the element of $\Prodsm{ D_{i} }{i \in I}$ 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~\tup{(2)}. Let $\gQ$ be a
complete congruence relation on
$\Prodsm{ D_{i} }{i \in I}$. 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}
\vct{i}{d} \equiv \vct{i}{c} \pod{\gQ},
\end{equation}
then $\gQ = \gi$.
\end{theorem}
\begin{proof}
Since
\begin{equation}\label{E:cong2}
\vct{i}{d} \equiv \vct{i}{c} \pod{\gQ},
\end{equation}
and $\gQ$ is a complete congruence relation, it follows
from condition~(3) that
\begin{align}\label{E:cong}
&\vct{i}{d} \equiv \notag\\
&\qq\q{\JJm{\vct{i}{c}}{d \leq c < 1}=1} \pod{\gQ}.
\end{align}
Let $j \in I$ for $j \neq i$, and let
$a \in D_{j}^{-}\). Meeting both sides of the congruence
\eqref{E:cong} with $\vct{j}{a}$, we obtain
\begin{align}\label{E:comp}
0 &= \vct{i}{d} \mm \vct{j}{a}\\
&\equiv \vct{j}{a}\pod{\gQ}. \notag
\end{align}
Using the completeness of $\gQ$ and \eqref{E:comp}, we get:
\begin{equation}\label{E:cong3}
0=\JJm{ \vct{j}{a} }{ a \in D_{j}^{-} } \equiv 1 \pod{\gQ},
\end{equation}
hence $\gQ = \gi$.
\end{proof}
\begin{theorem}\label{T:P*a}
Let $D_{i}$ for $i \in I$ be complete distributive
lattices satisfying
conditions~\tup{(2)} and \tup{(3)}. Then
$\Prodsm{ D_{i} }{i \in I}$ also satisfies
conditions~\tup{(2)} and \tup{(3)}.
\end{theorem}
\begin{proof}
Let $\gQ$ be a complete congruence on
$\Prodsm{ D_{i} }{i \in I}$. Let $i \in I$. Define
\begin{equation}\label{E:dihat}
\widehat{D}_{i} = \setm{ \vct{i}{d} }{ d \in D_{i}^{-} }
\uu \set{1}.
\end{equation}
Then $\widehat{D}_{i}$ is a complete sublattice of
$\Prodsm{ D_{i} }{i \in I}$, and $\widehat{D}_{i}$
is isomorphic to $D_{i}$. Let $\gQ_{i}$ be the
restriction of $\gQ$ to $\widehat{D}_{i}$. Since
$D_{i}$ is complete-simple, so is $\widehat{D}_{i}$,
hence $\gQ_{i}$ is $\go$ or $\gi$. If $\gQ_{i} = \go$
for all $i \in I$, then $\gQ = \go$.
If there is an $i \in I$, such that $\gQ_{i} = \gi$,
then $0 \equiv 1 \pod{\gQ}$, and hence $\gQ = \gi$.
\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}