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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>
 +)]