Mathematical Structures: Equivalence relations

Equivalence relations

http://mathcs.chapman.edu/structuresold/files/Equivalence_relations.pdf
%%run pdflatex

%


\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}}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\markboth{\today}{math.chapman.edu/structures}

\begin{document}
\textbf{\Large Equivalence relations}

\abbreviation{EqRel}

\begin{definition}
An \emph{equivalence relation} is a structure $\mathbf{X}=\langle X,\equiv\rangle$ such that $\equiv$ is a \emph{binary relation on $X$}
(i.e. $\equiv\ \subseteq X\times X$) that
is

reflexive:  $x\equiv x$

symmetric:  $x\equiv y\implies y\equiv x$

transitive: $x\equiv y\text{ and }y\equiv z\implies x\equiv z$

Remark: This is a template.
If you know something about this class, click on the Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.
\end{definition}

\begin{morphisms}
Let $\mathbf{X}$ and $\mathbf{Y}$ be equivalence relations. A morphism from $\mathbf{X}$ to $\mathbf{Y}$ is a function $h:A\rightarrow B$ that is a homomorphism:
$x\equiv^{\mathbf X} y\implies h(x)\equiv^{\mathbf Y}h(y)$
\end{morphisms}

\begin{definition}
An \emph{equivalence relation} is a \href{Preordered_sets.pdf}{qoset} that is \emph{symmetric}: $x\equiv y\implies y\equiv x$
\end{definition}

\begin{basic_results}
Equivalence relations are in 1-1 correspondence with \href{Partitions.pdf}{partitions}.
\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})

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

\begin{tabular}{|ll|}\hline
Classtype                       & quasivariety \\\hline
Quasiequational theory          & \\\hline
First-order theory              & \\\hline
Locally finite                  & yes\\\hline
Residual size                   & \\\hline
Congruence distributive         & no\\\hline
Congruence modular              & no\\\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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &2\\ f(3)= &3\\ f(4)= &5\\ f(5)= &7\\ \end{array}$\qquad
$\begin{array}{lr} f(6)= &11\\ f(7)= &15\\ f(8)= &22\\ f(9)= &30\\ f(10)= &42\\ \end{array}$

The number of (labelled) equivalance relations on an $n$ element set given by a sum of Stirlings formula (of the second kind).

\url{http://www.research.att.com/projects/OEIS?Anum=A000110}

The number of (nonisomorphic) equivalence relations is the number of partition patterns (= number of integer partitions).

\url{http://www.research.att.com/projects/OEIS?Anum=A000041}
\end{finite_members}

\begin{subclasses}\

\end{subclasses}

\begin{superclasses}\

\href{Preordered_sets.pdf}{Preordered sets} supervariety

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Lastname19xx}
F. Lastname, \emph{Title}, Journal, \textbf{1}, 23--45 \href{http://www.ams.org/mathscinet-getitem?mr=12a:08034}{MRreview}

\end{thebibliography}

\end{document}
%