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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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diff --git a/info/examples/mil3/intrart.tex b/info/examples/mil3/intrart.tex new file mode 100644 index 0000000000..724be01586 --- /dev/null +++ b/info/examples/mil3/intrart.tex @@ -0,0 +1,127 @@ +% 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, MN 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, 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} + |