Mathematical Structures: Directed partial orders

Directed partial orders

http://mathcs.chapman.edu/structuresold/files/Directed_partial_orders.pdf
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\begin{document}
\textbf{\Large Directed partial orders}

\abbreviation{DPO}
\begin{definition}
A \emph{directed partial order} is a poset $\mathbf{P}=\left\langle P,\leq \right\rangle$ that is \emph{directed}, i.e. every finite subset
of $P$ has an upper bound in $P$, or equivalently, $P\ne\emptyset$, $\forall xy\exists z (x\le z$ and $y\le z)$.
\end{definition}
\begin{morphisms}
Let $\mathbf{P}$ and $\mathbf{Q}$ be directed partial orders. A morphism from $\mathbf{P}$ to
$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is order preserving:

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

\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 & first-order\\\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 $n$.

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

\href{Directed_complete_partial_orders.pdf}{Directed complete partial orders}

\end{subclasses}
\begin{superclasses}\

\href{Partially_ordered_sets.pdf}{Partially ordered sets}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

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\end{document}
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