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+% 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{menuhin@ccw.uwinnebago.edu}
+\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} \ No newline at end of file