Mathematical Structures: History of Sets

# History of Sets

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 Revision 3 . . July 10, 2004 11:05 am by Jipsen Revision 2 . . (edit) May 30, 2003 11:17 pm by Peter Jipsen

Difference (from prior major revision) (no other diffs)

Changed: 1,94c1,94
 %%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 Sets} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Sets}{edit} \abbreviation{Set} \begin{definition} A \emph{set} is a structure $\mathbf{A}=\langle A\rangle$ with no operations or relations defined on $A$. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be sets. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$. \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & variety\\\hline Equational theory & decidable\\\hline Quasiequational theory & decidable\\\hline First-order theory & decidable\\\hline Locally finite & yes\\\hline Residual size & 2\\\hline Congruence distributive & no\\\hline Congruence modular & no\\\hline Congruence n-permutable & no\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline Congruence extension property & yes\\\hline Definable principal congruences & yes\\\hline Equationally def. pr. cong. & no\\\hline Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline Epimorphisms are surjective & yes\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(n)= &1\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{One-element_structures.pdf}{One-element structures} \end{subclasses} \begin{superclasses}\ \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %
 %%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 Sets} \quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Sets}{edit} \abbreviation{Set} \begin{definition} A \emph{set} is a structure $\mathbf{A}=\langle A\rangle$ with no operations or relations defined on $A$. \end{definition} \begin{morphisms} Let $\mathbf{A}$ and $\mathbf{B}$ be sets. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$. \end{morphisms} \begin{basic_results} \end{basic_results} \begin{examples} \begin{example} \end{example} \end{examples} \begin{table}[h] \begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description}) \begin{tabular}{|ll|}\hline Classtype & variety\\\hline Equational theory & decidable\\\hline Quasiequational theory & decidable\\\hline First-order theory & decidable\\\hline Locally finite & yes\\\hline Residual size & 2\\\hline Congruence distributive & no\\\hline Congruence modular & no\\\hline Congruence n-permutable & no\\\hline Congruence regular & no\\\hline Congruence uniform & no\\\hline Congruence extension property & yes\\\hline Definable principal congruences & yes\\\hline Equationally def. pr. cong. & no\\\hline Amalgamation property & yes\\\hline Strong amalgamation property & yes\\\hline Epimorphisms are surjective & yes\\\hline \end{tabular} \end{properties} \end{table} \begin{finite_members} $f(n)=$ number of members of size $n$. $\begin{array}{lr} f(n)= &1\\ \end{array}$ \end{finite_members} \hyperbaseurl{http://math.chapman.edu/structures/files/} \parskip0pt \begin{subclasses}\ \href{One-element_structures.pdf}{One-element structures} \end{subclasses} \begin{superclasses}\ \end{superclasses} \begin{thebibliography}{10} \bibitem{Ln19xx} \end{thebibliography} \end{document} %

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