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author | Karl Berry <karl@freefriends.org> | 2017-01-12 23:14:02 +0000 |
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committer | Karl Berry <karl@freefriends.org> | 2017-01-12 23:14:02 +0000 |
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diff --git a/Master/texmf-dist/doc/latex/math-into-latex-4/intropres.tex b/Master/texmf-dist/doc/latex/math-into-latex-4/intropres.tex new file mode 100644 index 00000000000..e05a03450e6 --- /dev/null +++ b/Master/texmf-dist/doc/latex/math-into-latex-4/intropres.tex @@ -0,0 +1,155 @@ +% Introductory presentation:intropres.tex + +\documentclass{foils} +\usepackage{amsmath} +\usepackage{amssymb,latexsym} +\usepackage{graphicx} + +\begin{document} +\title{A construction of complete-simple\\ + distributive lattices} +\author{George~A. Menuhin\\ + Computer Science Department\\ + University of Winnebago\\ + Winnebago, MN 53714} +\date{March 15, 2006} + +\maketitle +\begin{abstract} + In this presentation, we prove that there exist + \emph{complete-simple distributive lattices,} + that is, complete distributive lattices + with only two complete congruences. +\end{abstract} + +\foilhead{The result} +%\section{Introduction}\label{S:intro} +In this presentation, 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} + +\foilhead{The construction} +%\section{The $\Pi^{*}$ construction}\label{S:P*} +The following construction is crucial in the proof +of our theorem: + +\begin{Definition}\label{D:P*} +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} + +\foilhead{} + +\begin{figure}[hbt] +\centering\includegraphics[scale=2]{products} +\caption{The product construction illustrated}\label{Fi:products} +\end{figure} + +\foilhead{Notation} + + +If $i \in I$ and $d \in D_{i}^{-}$, then +\[ + \langle \dots, 0, \dots, d, \dots, 0, \dots \rangle +\] +is the element of\, $\Pi^{*} ( D_{i} \mid i \in I )$ whose +$i$-th component is $d$ and all the other components +are $0$. + + +See also Ernest~T. Moynahan, 1957. + +\foilhead{Second 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, d, \dots, 0, \dots \rangle \equiv + \langle \dots, c, \dots, 0, \dots \rangle + \pod{\Theta}, +\end{equation*} +then $\Theta = \iota$. +\end{Theorem} + +\foilhead{Verification} + +Since +\begin{equation*} +\langle \dots, d, \dots, 0, \dots \rangle \equiv +\langle \dots, c, \dots, 0, \dots \rangle +\pod{\Theta}, +\end{equation*} +and $\Theta$ is a complete congruence relation, +it follows from condition~(J) that modulo $\Theta$ +\begin{equation*}\label{E:cong} + \langle \dots, d, \dots, 0, \dots \rangle \equiv + \bigvee ( \langle \dots, c, \dots, 0, \dots \rangle + \mid d \leq c < 1 ). +\end{equation*} + +\foilhead{Verification completed} + +Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$. +Meeting both sides of the last congruence +with $\langle \dots, a, \dots, 0, \dots \rangle$, +we obtain that +\begin{equation*} + 0 = \langle \dots, a, \dots, 0, \dots \rangle + \pod{\Theta}, +\end{equation*} +Using the completeness of $\Theta$ and the previous equation, +we get: +\[ + 0 \equiv \bigvee ( \langle \dots, a, \dots, 0, + \dots \rangle \mid a \in D_{j}^{-} ) = 1 + \pod{\Theta}, +\] +hence $\Theta = \iota$. + +\foilhead{} + +\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{document} + |