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diff --git a/Master/texmf-dist/doc/latex/math-into-latex-4/quickbeamer1.tex b/Master/texmf-dist/doc/latex/math-into-latex-4/quickbeamer1.tex new file mode 100644 index 00000000000..aa60a13a2a1 --- /dev/null +++ b/Master/texmf-dist/doc/latex/math-into-latex-4/quickbeamer1.tex @@ -0,0 +1,173 @@ +%Introductory beamer presentation: quickbeamer1.tex + +\documentclass{beamer} +\usetheme{Berkeley} + +\begin{document} +\title{A construction of complete-simple\\ + distributive lattices} +\author{George~A. Menuhin} +\institute{Computer Science Department\\ + University of Winnebago\\ + Winnebago, MN 53714} +\date{March 15, 2006} + +\begin{frame} +\titlepage +\end{frame} + +\begin{frame} +\frametitle{Outline} + +\tableofcontents[pausesections] +\end{frame} + +\section{Introduction} + +\begin{frame} +\frametitle{Introduction} + +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} +\end{frame} + +\section{The $\Pi^{*}$ construction} + +\begin{frame} +\frametitle{The $\Pi^{*}$ construction} + +The following construction is crucial in the proof +of our Theorem: + +\begin{definition} +Let $D_{i}$, for $i \in I$, 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} +\end{frame} + +\begin{frame} +\frametitle{Illustrating the construction} + +\centering\includegraphics{products} +\end{frame} + +\begin{frame} +\frametitle{Notation} + +If $i \in I$ and $d \in D_{i}^{-}$, then +\[ + \langle \ldots, 0, \ldots, d, \ldots, 0, \ldots \rangle +\] +is the element of $\Pi^{*} ( D_{i} \mid i \in I )$ whose +$i$-th component is $d$ and all the other components +are $0$. + +See also Ernest~T. Moynahan, 1957. +\end{frame} + +\begin{frame} +\frametitle{The second result} + +Next we verify the following result: + +\begin{theorem} +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*} + \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv + \langle \ldots, c, \ldots, 0, \ldots \rangle + \pmod{\Theta}, +\end{equation*} +then $\Theta = \iota$. +\end{theorem} + +\end{frame} + +\begin{frame} +\frametitle{Starting the proof} + +Since +\begin{equation*} +\langle \ldots, d, \ldots, 0, \ldots \rangle \equiv +\langle \ldots, c, \ldots, 0, \ldots \rangle +\pmod{\Theta}, +\end{equation*} +and $\Theta$ is a complete congruence relation, +it follows from condition~(J) that +\begin{equation*} + \langle \ldots, d, \ldots, 0, \ldots \rangle \equiv + \bigvee ( \langle \ldots, c, \ldots, 0, \ldots \rangle + \mid d \leq c < 1 ) \pmod{\Theta}. +\end{equation*} +\end{frame} + +\begin{frame} +\frametitle{Completing the proof} + +Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$. +Meeting both sides of the congruence +with $\langle \ldots, a, \ldots, 0, \ldots \rangle$, +we obtain that +\begin{equation*} + 0 = \langle \ldots, a, \ldots, 0, \ldots \rangle + \pmod{\Theta}, +\end{equation*} +Using the completeness of $\Theta$ and the penultimate equation, +we get: +\[ + 0 \equiv \bigvee ( \langle \ldots, a, \ldots, 0, + \ldots \rangle \mid a \in D_{j}^{-} ) = 1 + \pmod{\Theta}, +\] +hence $\Theta = \iota$. +\end{frame} + +\begin{frame} +\frametitle{References} + +\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. \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{frame} +\end{document} + |