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diff --git a/info/examples/Practical_LaTeX/firstdocument.tex b/info/examples/Practical_LaTeX/firstdocument.tex new file mode 100644 index 0000000000..d5851a08c7 --- /dev/null +++ b/info/examples/Practical_LaTeX/firstdocument.tex @@ -0,0 +1,78 @@ +%First document, firstdocument.tex +\documentclass{amsart} +\usepackage{amssymb,latexsym} + +\newtheorem{theorem}{Theorem} + +\begin{document} +\title{A technical result\\ for congruences of finite lattices} +\author{G. Gr\"atzer} +\address{Department of Mathematics\\ + University of Manitoba\\ + Winnipeg, MB R3T 2N2\\ + Canada} +\email[G. Gr\"atzer]{gratzer@me.com} +\urladdr[G. Gr\"atzer]{http://tinyurl.com/gratzerhomepage} +\date{March 21, 2014} +\subjclass[2010]{Primary: 06B10.} +\keywords{finite lattice, congruence.} +\begin{abstract} +We present a technical result for congruences on finite lattices. +\end{abstract} +\maketitle + +\section{Introduction}\label{S:Introduction}%Section~\label{S:Introduction} +In some recent research, G. Cz\'edli +and I, see \cite{gC13} and \cite{gG14}, spent quite an effort +in proving that some equivalence relations +on a planar semimodular lattice are congruences. +The number of cases we had to consider +was dramatically cut by the following result. + +\begin{theorem}\label{T:technical}%Theorem~\ref{T:technical} +Let $L$ be a finite lattice. +Let $\delta$ be an equivalence relation on $L$ +with intervals as equivalence classes. +Then $\delta$ is a congruence relation if{}f +the following condition and its dual hold: +\begin{equation}\label{E:cover}%\eqref{E:cover} +\text{If $x$ is covered by $y,z \in L$ +and $x \equiv y \pmod{\delta}$, +then $z \equiv y + z \pmod{\delta}$.}\tag{C${}_{+}$} +\end{equation} +\end{theorem} + +\section{The proof}\label{Proof}%Section~\label{S:Proof} +We prove the join-substitution property: +if $x \leq y$ and $x \equiv y \pmod{\delta}$, then +\begin{equation}\label{E:Cjoin}%\eqref{E:Cjoin} +x + z \equiv y + z \pmod{\delta}. +\end{equation} +Let $U = [x, y+ z]$. +We induct on length\,$U$, the length of $U$. + +Let $I=[y_1,y+ z]$ and $J=[z_1,y+ z]$. +Then length\,$I$ and length\,$J < $ length\,$U$. +Hence, the induction hypothesis applies to $I$ +and $\delta\rceil I$, and we obtain that +$w \equiv y+ w \pmod{\delta}$. +By the transitivity of $\delta$, we conclude that +\begin{equation}\label{E:three}%\eqref{E:three} +z_1 \equiv y+ w \pmod{\delta}. +\end{equation} +Therefore, applying the induction hypothesis to $J$ +and $\delta \rceil J$, we conclude \eqref{E:Cjoin}. + +\begin{thebibliography}{9} +\bibitem{gC13}%G. Cz\'edli~\cite{gC13} +G. Cz\'edli, +\emph{Patch extensions and trajectory colorings of slim +rectangular lattices.} +Algebra Universalis \textbf{88} (2013), 255--280. + +\bibitem{gG14}%G. Gr\"atzer \cite{GS13} +G. Gr\"atzer, +\emph{Congruences of fork extensions of lattices.} +Acta Sci. Math. (Szeged), \textbf{57} (2014), 417--434. +\end{thebibliography} +\end{document}
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