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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