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author | Karl Berry <karl@freefriends.org> | 2011-03-07 17:05:34 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2011-03-07 17:05:34 +0000 |
commit | e3b8dc2318b24a320bd0f057f5be50b89ef2241a (patch) | |
tree | 179f080e3868a391a1d16aaf1cdbd13eae9922ba /Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex | |
parent | fa94e399e78544fb77abd60e1b0090b6248d6d1d (diff) |
rm math-into-latex, noinfo license
git-svn-id: svn://tug.org/texlive/trunk@21625 c570f23f-e606-0410-a88d-b1316a301751
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diff --git a/Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex b/Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex deleted file mode 100644 index dbfa149f7af..00000000000 --- a/Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex +++ /dev/null @@ -1,133 +0,0 @@ -% Introductory sample article: intrart.tex -% Typeset with LaTeX format - -\documentclass{article} -\usepackage{amsmath,amssymb} -\newtheorem{theorem}{Theorem} -\newtheorem{definition}{Definition} -\newtheorem{notation}{Notation} - -\begin{document} -\title{A construction of complete-simple\\ - distributive lattices} -\author{George~A. Menuhin\thanks{Research supported - by the NSF under grant number~23466.}\\ - Computer Science Department\\ - Winnebago, Minnesota 23714\\ - menuhin@ccw.uwinnebago.edu} -\date{March 15, 1995} -\maketitle - -\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} - -\section{Introduction} \label{S:intro} -In this note we prove the following result: - -\begin{theorem} - There exists an infinite complete distributive lattice $K$ - with only the two trivial complete congruence relations. -\end{theorem} - -\section{The $\Pi^{*}$ construction} \label{S:P*} -The following construction is crucial in our proof of our Theorem: - -\begin{definition} \label{D:P*} - Let $D_{i}$, $i \in I$, be complete distributive - lattices satisfying condition~\textup{(J)}. Their - $\Pi^{*}$ product is defined as follows: - \[ - \Pi^{*} ( D_{i} \mid i \in I ) = - \Pi ( D_{i}^{-} \mid i \in I ) + 1; - \] - that is, $\Pi^{*} ( D_{i} \mid i \in I )$ is - $\Pi ( D_{i}^{-} \mid i \in I )$ with a new unit element. -\end{definition} - -\begin{notation} - If $i \in I$ and $d \in D_{i}^{-}$, then - \[ - \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, - \dots \rangle - \] - is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ 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 the following result: - -\begin{theorem} \label{T:P*} - Let $D_{i}$, $i \in I$, be complete distributive - lattices satisfying condition~\textup{(J)}. Let $\Theta$ - be a complete congruence relation on - $\Pi^{*} ( D_{i} \mid i \in I )$. - 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} - \langle \dots, 0, \dots,\overset{i}{d}, - \dots, 0, \dots \rangle \equiv \langle \dots, 0, \dots, - \overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta}, - \end{equation} - then $\Theta = \iota$. -\end{theorem} - -\emph{Proof.} Since -\begin{equation} \label{E:cong2} - \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, - \dots \rangle \equiv \langle \dots, 0, \dots, - \overset{i}{c}, \dots, 0, \dots \rangle \pmod{\Theta}, -\end{equation} -and $\Theta$ is a complete congruence relation, it follows -from condition~(C) that -\begin{align} \label{E:cong} - & \langle \dots, \overset{i}{d}, \dots, 0, - \dots \rangle \equiv\\ - &\qquad \qquad \quad \bigvee ( \langle \dots, 0, \dots, - \overset{i}{c}, \dots, 0, \dots \rangle \mid d \leq c < 1 ) - \equiv 1 \pmod{\Theta}. \notag -\end{align} - -Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$. -Meeting both sides of the congruence \eqref{E:cong2} with -$\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, -\dots \rangle$, we obtain - -\begin{align} \label{E:comp} - 0 = & \langle \dots, 0, \dots, \overset{i}{d}, \dots, 0, \dots - \rangle \wedge \langle \dots, 0, \dots, \overset{j}{a}, - \dots, 0, \dots \rangle \equiv\\ - &\langle \dots, 0, \dots, \overset{j}{a}, \dots, 0, \dots - \rangle \pmod{\Theta}, \notag -\end{align} -Using the completeness of $\Theta$ and \eqref{E:comp}, -we get: -\[ - 0 \equiv \bigvee ( \langle \dots, 0, \dots, \overset{j}{a}, - \dots, 0, \dots \rangle \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta}, -\] -hence $\Theta = \iota$. - -\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} - Ernest~T. Moynahan, \emph{Ideals and congruence relations in - lattices.~II}, Magyar Tud. Akad. Mat. Fiz. Oszt. K\"{o}zl. \textbf{9} - (1957), 417--434. -\end{thebibliography} - -\end{document} - |