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+% Introductory legacy sample article: legacy-article.tex
+
+\documentclass{article}
+\usepackage{latexsym}
+\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, MN 23714\\
+ menuhin@cc.uwinnebago.edu}
+\date{March 15, 2006}
+\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 the proof of our Theorem:
+
+\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
+ \[
+ \langle \ldots, 0, \ldots, d^i, \ldots, 0, \ldots \rangle
+ \]
+ 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 \ldots, d, \ldots, 0, \ldots \rangle \equiv
+ \langle \ldots, c, \ldots, 0, \ldots \rangle \pmod{\Theta},
+ \end{equation}
+ then $\Theta = \iota$.
+\end{theorem}
+
+\emph{Proof.} Since
+\begin{equation}\label{E:cong2}
+ \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv
+ \langle \ldots, c, \ldots, 0, \ldots \rangle \pmod{\Theta},
+\end{equation}
+and $\Theta$ is a complete congruence relation, it follows
+from condition~(J) that
+\begin{equation}\label{E:cong}
+ \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv
+ \bigvee ( \langle \ldots, c, \ldots, 0, \ldots \rangle
+ \mid d \leq c < 1 ) \pmod{\Theta}.
+\end{equation}
+
+Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$.
+Meeting both sides of the congruence (\ref{E:cong2}) with
+$\langle \ldots, a, \ldots, 0, \ldots \rangle$, we obtain that
+\begin{equation}\label{E:comp}
+ 0 = \langle \ldots, a, \ldots, 0, \ldots \rangle \pmod{\Theta},
+\end{equation}
+Using the completeness of $\Theta$ and (\ref{E:comp}),
+we get:
+\[
+ 0 \equiv \bigvee ( \langle \ldots, a, \ldots, 0, \ldots
+ \rangle \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta},
+\]
+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, 1968.
+ \bibitem{eM57}
+ Ernest~T. Moynahan,
+ \emph{On a problem of M. 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}
+