# Differences

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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> | ||

+ | )] |

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