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-% Sample file: sampartu.tex
-% The sample article for the amsart document class
-% with user-defined commands and environments
-% Typeset with LaTeX format
-
-\documentclass{amsart}
-\usepackage{amssymb,latexsym}
-\usepackage{lattice}
-
-\theoremstyle{plain}
-\newtheorem{theorem}{Theorem}
-\newtheorem{corollary}{Corollary}
-\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.edu/homepages/menuhin/}
-\thanks{Research supported by the NSF under grant number~23466.}
-\keywords{Complete lattice, distributive lattice, complete
- congruence, congruence lattice}
-\subjclass[2000]{Primary: 06B10; Secondary: 06D05}
-\date{March 15, 1999}
-
-\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{named}{Main Theorem}
- There exists an infinite complete distributive lattice
- $K$ with only the two trivial complete congruence relations.
-\end{named}
-
-\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{enumeratei}
- \item $u$ is meet-irreducible but $u$ is \emph{not}
- completely meet-irreducible;\label{m-i}
- \item $v$ is join-irreducible but $v$ is \emph{not}
- completely join-irreducible;\label{j-i}
- \item $[u, v]$ is a complete-simple lattice.\label{c-s}
- \end{enumeratei}
-\end{definition}
-
-Now we prove the following result:
-
-\begin{lemma}\label{L:ds}
- Let $D$ be a complete distributive lattice satisfying
- conditions \eqref{m-i} and~\eqref{j-i}.
- Then $\Ds$ is a sublattice of $D^{2}$; hence $\Ds$ is
- a lattice, and $\Ds$ is a complete distributive lattice
- satisfying conditions \eqref{m-i} and~\eqref{j-i}.
-\end{lemma}
-
-\begin{proof}
- By conditions~\eqref{m-i} and \eqref{j-i}, $\Ds$ 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 \eqref{m-i}
- and~\eqref{j-i} 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}
- \con \vv<1, d>=\vv<1, 1>(\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~\eqref{j-i}. 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~\eqref{j-i}. 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}
- \con\vct{i}{d}=\vct{i}{c}(\gQ),
- \end{equation}
- then $\gQ = \gi$.
-\end{theorem}
-
-\begin{proof}
- Since
- \begin{equation}\label{E:cong2}
- \con\vct{i}{d}=\vct{i}{c}(\gQ),
- \end{equation}
- and $\gQ$ is a complete congruence relation, it follows
- from condition~\eqref{c-s} that
- \begin{equation}\label{E:cong}
- \begin{split}
- &\langle \dots, \overset{i}{d}, \dots, 0,
- \dots \rangle\\
- &\equiv \bigvee ( \langle \dots, 0, \dots,
- \overset{i}{c},\dots, 0,\dots \rangle \mid d \leq c < 1)
- \equiv 1 \pmod{\Theta}.
- \end{split}
- \end{equation}
-
- 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{equation}\label{E:comp}
- \begin{split}
- 0 &= \vct{i}{d} \mm \vct{j}{a}\\
- &\equiv \vct{j}{a}\pod{\gQ}.
- \end{split}
- \end{equation}
- Using the completeness of $\gQ$ and \eqref{E:comp}, we get:
- \begin{equation}\label{E:cong3}
- \con{0=\JJm{ \vct{j}{a} }{ a \in D_{j}^{-} }}={1}(\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 \eqref{j-i} and~\eqref{c-s}. Then
- $\Prodsm{ D_{i} }{i \in I}$ also satisfies
- conditions~\eqref{j-i} and \eqref{c-s}.
-\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 $\con0=1(\gQ)$, 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, 1968.
-
- \bibitem{eM57}
- Ernest~T. Moynahan, \emph{On a problem of M. Stone},
- Acta Math. Acad. Sci. Hungar. \tbf{8} (1957), 455--460.
-
- \bibitem{eM57a}
- \bysame, \emph{Ideals and congruence relations in
- lattices}.~II, Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl.
- \tbf{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