Mathematical Structures: Euclidean domains

Euclidean domains

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Changed: 1,106c1,106
 %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Euclidean Domains} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Euclidean_Domains}{edit} \abbreviation{EucDom} \begin{definition} A \emph{Euclidean domain} is an \href{Integral_domains.pdf}{integral domains} $\langle D,+,-,0,\cdot,1\rangle$ together with a function $d:D\setminus\{0\} \to\mathbf{N}$ such that $\forall a,b\ (a\ne 0$, $b\neq 0 \implies d(a)\le d(ab))$ $\forall a,b \exists q,r\ (a=b\cdot q+r$, $(r=0 \mbox{or} d(r)  %%run pdflatex % \documentclass[12pt]{amsart} \usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref} \parindent=0pt \parskip=5pt \addtolength{\oddsidemargin}{-.5in} \addtolength{\evensidemargin}{-.5in} \addtolength{\textwidth}{1in} \theoremstyle{definition} \newtheorem{definition}{Definition} \newtheorem*{morphisms}{Morphisms} \newtheorem*{basic_results}{Basic Results} \newtheorem*{examples}{Examples} \newtheorem{example}{} \newtheorem*{properties}{Properties} \newtheorem*{finite_members}{Finite Members} \newtheorem*{subclasses}{Subclasses} \newtheorem*{superclasses}{Superclasses} \newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}} \pagestyle{myheadings}\thispagestyle{myheadings} \markboth{\today}{math.chapman.edu/structures} \begin{document} \textbf{\Large Euclidean Domains} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Euclidean_Domains}{edit} \abbreviation{EucDom} \begin{definition} A \emph{Euclidean domain} is an \href{Integral_domains.pdf}{integral domains}$\langle D,+,-,0,\cdot,1\rangle$together with a function$d:D\setminus\{0\} \to\mathbf{N}$such that$\forall a,b\ (a\ne 0$,$b\neq 0 \implies d(a)\le d(ab))\forall a,b \exists q,r\ (a=b\cdot q+r$,$(r=0 \mbox{or} d(r)

http://mathcs.chapman.edu/structuresold/files/Euclidean_domains.pdf
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\documentclass[12pt]{amsart}
\usepackage[pdfpagemode=Fullscreen,pdfstartview=FitBH]{hyperref}
\parindent=0pt
\parskip=5pt
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\newtheorem*{morphisms}{Morphisms}
\newtheorem*{basic_results}{Basic Results}
\newtheorem*{examples}{Examples}
\newtheorem{example}{}
\newtheorem*{properties}{Properties}
\newtheorem*{finite_members}{Finite Members}
\newtheorem*{subclasses}{Subclasses}
\newtheorem*{superclasses}{Superclasses}
\newcommand{\abbreviation}[1]{\textbf{Abbreviation: #1}}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Euclidean Domains}

\abbreviation{EucDom}
\begin{definition}
A \emph{Euclidean domain} is an \href{Integral_domains.pdf}{integral domains} $\langle D,+,-,0,\cdot,1\rangle$ together with a function $d:D\setminus\{0\} \to\mathbf{N}$ such that

$\forall a,b\ (a\ne 0$, $b\neq 0 \implies d(a)\le d(ab))$

$\forall a,b \exists q,r\ (a=b\cdot q+r$, $(r=0 \mbox{or} d(r)<d(b)))$
\end{definition}

\begin{morphisms}

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\langle\mathbb{Z},+,-,0,\cdot,1,d\rangle$, the ring of integers with addition, subtraction, zero, and multiplication is a Euclidean domain with $d(a)=|a|$.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & first-order\\\hline
Equational theory & \\\hline
Quasiequational theory & \\\hline
First-order theory & \\\hline
Locally finite & \\\hline
Residual size & \\\hline
Congruence distributive & \\\hline
Congruence modular & \\\hline
Congruence n-permutable & \\\hline
Congruence regular & \\\hline
Congruence uniform & \\\hline
Congruence extension property & \\\hline
Definable principal congruences & \\\hline
Equationally def. pr. cong. & \\\hline
Amalgamation property & \\\hline
Strong amalgamation property & \\\hline
Epimorphisms are surjective & \\\hline
\end{tabular}
\end{properties}
\end{table}
\begin{finite_members} $f(n)=$ number of members of size $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &1\\ f(5)= &1\\ f(6)= &0\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Fields.pdf}{Fields}

\end{subclasses}
\begin{superclasses}\

\href{Principal_Ideal_Domains.pdf}{Principal Ideal Domains}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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