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diff --git a/Master/texmf-dist/doc/latex/mil3/sampartu.tex b/Master/texmf-dist/doc/latex/mil3/sampartu.tex new file mode 100644 index 00000000000..41ef8bfe5e0 --- /dev/null +++ b/Master/texmf-dist/doc/latex/mil3/sampartu.tex @@ -0,0 +1,259 @@ +% 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}
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