Mathematical Structures: Directed complete partial orders

# Directed complete partial orders

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http://mathcs.chapman.edu/structuresold/files/Directed_complete_partial_orders.pdf
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\newtheorem*{morphisms}{Morphisms}
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\begin{document}
\textbf{\Large Directed complete partial orders}

\abbreviation{DCPO}
\begin{definition}
A \emph{directed complete partial order} is a poset $\mathbf{P}=\left\langle P,\leq \right\rangle$
such that every directed subset of $P$ has a least upper bound:
$\forall D\subseteq P\ (D\ne\emptyset\mbox{and}\forall x,y\in D\ \exists z\in D (x,y\le z)\implies \exists z\in P(z=\bigvee D))$.
\end{definition}
\begin{morphisms}
Let $\mathbf{P}$ and $\mathbf{Q}$ be directed complete partial orders. A morphism from $\mathbf{P}$ to
$\mathbf{Q}$ is a function $f:P\rightarrow Q$ that is \emph{Scott-continuous}, which means that $f$ preserves all directed joins:

$z=\bigvee D\implies f(z)= \bigvee f[D]$

\end{morphisms}
\begin{basic_results}
\end{basic_results}
\begin{examples}
\begin{example}
$\left\langle \mathbb{R},\leq \right\rangle$, the real numbers with the standard order.
\end{example}
\begin{example}
$\left\langle P(S),\subseteq \right\rangle$, the collection of subsets of a
sets $S$, ordered by inclusion.
\end{example}
\end{examples}
\begin{table}[h]
\begin{properties} (\href{http://math.chapman.edu/cgi-bin/structures?Properties}{description})

\begin{tabular}{|ll|}\hline
Classtype & second-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 $n$.

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$
\end{finite_members}
\hyperbaseurl{http://math.chapman.edu/structures/files/}
\parskip0pt
\begin{subclasses}\

\href{Complete_semilattices.pdf}{Complete semilattices}

\end{subclasses}
\begin{superclasses}\

\href{Directed_partial_orders.pdf}{Directed partial orders}

\end{superclasses}

\begin{thebibliography}{10}

\bibitem{Ln19xx}

\end{thebibliography}

\end{document}
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