Preordered sets

http://mathcs.chapman.edu/structuresold/files/Preordered_sets.pdf

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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Preordered sets}
\quad\href{http://math.chapman.edu/cgi-bin/structures?action=edit;id=Preordered_sets}{edit}

\abbreviation{Qoset}

\begin{definition}
A \emph{preordered set} (also called a \emph{quasi-ordered set} or \emph{qoset} for short) is a structure $\mathbf{P}=\langle P,\preceq\rangle$ 
such that $P$ is a set and $\preceq $ is a binary relation on $P$ that is

reflexive:  $x\preceq x$ and

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

Remark: 
\end{definition}

\begin{morphisms}
Let $\mathbf{P}$ and $\mathbf{Q}$ be qosets. A morphism from $\mathbf{P}$ to 
$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is preorder-preserving: 

$x\preceq y\implies f(x)\preceq f(y)$
\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 Horn class\\\hline
Universal theory & Decidable\\\hline
First-order theory & Undecidable\\\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 $n$.

$\begin{array}{lr}
f(1)= &1\\
f(2)= &2\\
f(3)= &\\
f(4)= &\\
f(5)= &\\
f(6)= &\\
f(7)= &\\
\end{array}$
\end{finite_members}
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\begin{subclasses}\ 

\href{Posets.pdf}{Posets} 

\href{Connected_qosets.pdf}{Connected qosets} 

\end{subclasses}
\begin{superclasses}\ 

\href{Binary_relational_structures.pdf}{Binary relational structures} 

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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