diff options
Diffstat (limited to 'Master/texmf-doc/doc/english/firststeps/intrart.tex')
-rw-r--r-- | Master/texmf-doc/doc/english/firststeps/intrart.tex | 127 |
1 files changed, 0 insertions, 127 deletions
diff --git a/Master/texmf-doc/doc/english/firststeps/intrart.tex b/Master/texmf-doc/doc/english/firststeps/intrart.tex deleted file mode 100644 index 1f35c477230..00000000000 --- a/Master/texmf-doc/doc/english/firststeps/intrart.tex +++ /dev/null @@ -1,127 +0,0 @@ -% Introductory sample article: intrart.tex -% Typeset with LaTeX format - -\documentclass{article} -\usepackage{latexsym} -\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@cc.uwinnebago.edu} -\date{March 15, 1999} -\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 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} - -\begin{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 $0$. -\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 \ldots, d, \ldots, 0, \ldots \rangle \equiv - \langle \ldots, c, \ldots, 0, \ldots \rangle \pmod{\Theta}, - \end{equation} - then $\Theta = \iota$. -\end{theorem} - -\emph{Proof.} Since -\begin{equation}\label{E:cong2} - \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}\label{E:cong} - \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} - -Let $j \in I$, $j \neq i$, and let $a \in D_{j}^{-}$. -Meeting both sides of the congruence (\ref{E:cong2}) with -$\langle \ldots, a, \ldots, 0, \ldots \rangle$, we obtain that -\begin{equation}\label{E:comp} - 0 = \langle \ldots, a, \ldots, 0, \ldots \rangle \pmod{\Theta}, -\end{equation} -Using the completeness of $\Theta$ and (\ref{E:comp}), -we get: -\[ - 0 \equiv \bigvee ( \langle \ldots, a, \ldots, 0, \ldots \rangle - \mid a \in D_{j}^{-} ) = 1 \pmod{\Theta}, -\] -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-Melbourne, 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} - |