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-% Introductory sample article: intrart.tex
-% Typeset with LaTeX format
-
-\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, Minnesota 23714\\
- menuhin@cc.uwinnebago.edu}
-\date{March 15, 1999}
-\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
- \[
- \langle \ldots, 0, \ldots, d, \ldots, 0, \ldots \rangle
- \]
- 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-Toronto-London-Melbourne, 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}
-