summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/latex/math-into-latex-4/intrart.tex
diff options
context:
space:
mode:
Diffstat (limited to 'Master/texmf-dist/doc/latex/math-into-latex-4/intrart.tex')
-rw-r--r--Master/texmf-dist/doc/latex/math-into-latex-4/intrart.tex150
1 files changed, 150 insertions, 0 deletions
diff --git a/Master/texmf-dist/doc/latex/math-into-latex-4/intrart.tex b/Master/texmf-dist/doc/latex/math-into-latex-4/intrart.tex
new file mode 100644
index 00000000000..6ff0a310f6b
--- /dev/null
+++ b/Master/texmf-dist/doc/latex/math-into-latex-4/intrart.tex
@@ -0,0 +1,150 @@
+% Introductory sample article: intrart.tex
+
+\documentclass{amsart}
+\usepackage{amssymb,latexsym}
+\usepackage{graphicx}
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}{Lemma}
+\newtheorem{definition}{Definition}
+\newtheorem{notation}{Notation}
+
+\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}
+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 (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. 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 \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 $\Theta = \iota$.
+\end{proof}
+
+\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}
+