Mathematical Structures: Chains

# Chains

http://mathcs.chapman.edu/structuresold/files/Chains.pdf
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\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}
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\begin{document}
\textbf{\Large Chains}

\begin{definition}
A \emph{chain} is a \href{Partially_ordered_sets.pdf}{partially ordered set} $\mathbf{C}=\langle C,\le\rangle$ such that

$\le$ is a total order:  $x\le y \mbox{ or } y\le x$

Remark:

\end{definition}
\begin{morphisms}
Let $\mathbf{C}$ and $\mathbf{D}$ be chains. A morphism from $\mathbf{C}$ to $\mathbf{D}$ is a function $h:C\rightarrow D$ that is a orderpreserving:

$x\le y\implies h(x)\le h(y)$

\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 & Universal\\\hline
Quasiequational theory & \\\hline
First-order theory & \\\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 $n$.

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

\href{Well-ordered_chains.pdf}{Well-ordered chains}

\href{Dense_linear_orders.pdf}{Dense linear orders}

\end{subclasses}
\begin{superclasses}\

\href{Trees.pdf}{Trees}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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