# Differences

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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> | ||

+ | )] |

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