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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>
 +)]