summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex
diff options
context:
space:
mode:
authorKarl Berry <karl@freefriends.org>2011-03-07 17:05:34 +0000
committerKarl Berry <karl@freefriends.org>2011-03-07 17:05:34 +0000
commite3b8dc2318b24a320bd0f057f5be50b89ef2241a (patch)
tree179f080e3868a391a1d16aaf1cdbd13eae9922ba /Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex
parentfa94e399e78544fb77abd60e1b0090b6248d6d1d (diff)
rm math-into-latex, noinfo license
git-svn-id: svn://tug.org/texlive/trunk@21625 c570f23f-e606-0410-a88d-b1316a301751
Diffstat (limited to 'Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex')
-rw-r--r--Master/texmf-dist/doc/latex/math-into-latex/templates/intrart.tex133
1 files changed, 0 insertions, 133 deletions
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}
-