# Differences

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directed_partial_orders [2010/07/29 18:30] (current)
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+=====Directed partial orders=====
+Abbreviation: **DPO**
+====Definition====
+A \emph{directed partial order} is a poset $\mathbf{P}=\langle P,\leq \rangle$ 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 $y\le z)$.
+==Morphisms==
+Let $\mathbf{P}$ and $\mathbf{Q}$ be directed partial orders. A morphism from $\mathbf{P}$ to
+$\mathbf{Q}$ is a function $f:Parrow Q$ that is order preserving:
+
+$x\le y\Longrightarrow f(x)\le f(y)$
+
+====Examples====
+Example 1:
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |first-order |
+^[[Amalgamation property]]  | |
+^[[Strong amalgamation property]]  | |
+^[[Epimorphisms are surjective]]  | |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &2\\ +f(4)= &\\ +f(5)= &\\ +f(6)= &\\ +\end{array}$
+
+====Subclasses====
+[[Directed complete partial orders]]
+
+====Superclasses====
+[[Partially ordered sets]]
+
+
+====References====
+
+[(Ln19xx>
+)]