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diff --git a/Master/texmf-doc/doc/english/math-into-latex/templates/intrart.tex b/Master/texmf-doc/doc/english/math-into-latex/templates/intrart.tex new file mode 100644 index 00000000000..dbfa149f7af --- /dev/null +++ b/Master/texmf-doc/doc/english/math-into-latex/templates/intrart.tex @@ -0,0 +1,133 @@ +% 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} + |