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diff --git a/Master/texmf-doc/doc/english/firststeps/sampart2.tex b/Master/texmf-doc/doc/english/firststeps/sampart2.tex deleted file mode 100644 index 27a24e3a882..00000000000 --- a/Master/texmf-doc/doc/english/firststeps/sampart2.tex +++ /dev/null @@ -1,252 +0,0 @@ -% 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}
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