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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>
		+	)]