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diff --git a/Master/texmf-dist/doc/latex/math-into-latex-4/intrarti.tex b/Master/texmf-dist/doc/latex/math-into-latex-4/intrarti.tex new file mode 100644 index 00000000000..3add8c997b4 --- /dev/null +++ b/Master/texmf-dist/doc/latex/math-into-latex-4/intrarti.tex @@ -0,0 +1,157 @@ +% Introductory sample article with index entries: intrarti.tex + +\documentclass{amsart} +\usepackage{amssymb,latexsym} +\usepackage{graphicx} +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{definition}{Definition} +\newtheorem{notation}{Notation} + +\makeindex + +\begin{document} +\title{A construction of complete-simple\\ + distributive lattices} +\author{George~A. Menuhin} +\address{Computer Science Department\\ + University of Winnebago\\ + Winnebago, MN 53714} +\date{March 15, 2006} + +\begin{abstract} + In this note, we prove that there exist + \emph{complete-simple distributive lattices,} + that is, complete distributive lattices + with only two complete congruences. +\end{abstract} + +\maketitle + +\section{Introduction}\label{S:intro} +In this note we prove the following result: + +\begin{theorem}\index{Main 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*} +\index{pistar@$\Pi^{*}$ construction}% +\index{Main Theorem!exposition|(}% +The following construction is crucial in the proof +of our Theorem (see Figure~\ref{Fi:products}): + +\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} + +\begin{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 $0$. +\end{notation} + +See also Ernest~T. +\index{Moynahan, Ernest~T.}% +Moynahan~\cite{eM57a}. + +Next we verify the following result: +\index{lattice|textbf}% +\index{lattice!distributive}% +\index{lattice!distributive!complete}% +\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} + +\begin{figure}[hbt] +\centering\includegraphics{products} +\caption{}\label{Fi:products} +\end{figure} + +\begin{proof} +Since +\begin{equation}\label{E:cong2} +\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 +\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 ) \pod{\Theta}. +\end{equation} + +Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$. +Meeting both sides of the congruence \eqref{E:cong2} +with $\langle \dots, a, \dots, 0, \dots \rangle$, +we obtain that +\begin{equation}\label{E:comp} + 0 = \langle \dots, a, \dots, 0, \dots \rangle + \pod{\Theta}, +\end{equation} +Using the completeness of $\Theta$ and \eqref{E:comp}, +we get: +\[ + 0 \equiv \bigvee ( \langle \dots, a, \dots, 0, + \dots \rangle \mid a \in D_{j}^{-} ) = 1 + \pod{\Theta}, +\] +hence $\Theta = \iota$. +\index{Main Theorem!exposition|)} +\end{proof} + +\begin{thebibliography}{9} + \bibitem{sF90}\index{Foo, Soo-Key}% + Soo-Key Foo, + \emph{Lattice Constructions}, + Ph.D. thesis, + University of Winnebago, Winnebago, MN, December, 1990. + + \bibitem{gM68}\index{Menuhin, George~A.}% + George~A. Menuhin, + \emph{Universal Algebra}. + D.~Van Nostrand, Princeton, 1968. + + \bibitem{eM57}\index{Moynahan, Ernest~T.}% + Ernest~T. Moynahan, + \emph{On a problem of M. Stone}, + Acta Math. Acad. Sci. Hungar. \textbf{8} (1957), 455--460. + + \bibitem{eM57a}\index{Moynahan, Ernest~T.}% + 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} + +\printindex +\end{document} + |