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+% Introductory sample article: intrart.tex
+% Typeset with LaTeX format
+
+\documentclass{article}
+\usepackage{amsmath,amssymb}
+\newtheorem{theorem}{Theorem}
+\newtheorem{definition}{Definition}
+\newtheorem{notation}{Notation}
+
+\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\\
+ menuhin@ccw.uwinnebago.edu}
+\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}
+