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directed_complete_partial_orders [2010/07/29 18:30] (current)
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+=====Directed complete partial orders=====
+Abbreviation: **DCPO**
+====Definition====
+A \emph{directed complete partial order} is a poset $\mathbf{P}=\langle P,\leq \rangle$
+such that every directed subset of $P$ has a least upper bound:
+$\forall D\subseteq P\ (D\ne\emptyset\mbox{and}\forall x,y\in D\ \exists z\in D +(x,y\le z)\Longrightarrow \exists z\in P(z=\bigvee D))$.
+==Morphisms==
+Let $\mathbf{P}$ and $\mathbf{Q}$ be directed complete partial orders. A morphism from $\mathbf{P}$ to
+$\mathbf{Q}$ is a function $f:Parrow Q$ that is \emph{Scott-continuous}, which means that $f$ preserves all directed joins:
+
+$z=\bigvee D\Longrightarrow f(z)= \bigvee f[D]$
+
+====Examples====
+Example 1: $\langle \mathbb{R},\leq \rangle$, the real numbers with the standard order.
+Example 1: $\langle P(S),\subseteq \rangle$, the collection of subsets of a
+sets $S$, ordered by inclusion.
+
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |second-order |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &\\ +f(3)= &\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====Subclasses====
+[[Complete semilattices]]
+
+====Superclasses====
+[[Directed partial orders]]
+
+
+====References====
+
+[(Ln19xx>
+)]