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