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\begin{document}

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\section*{dcpo}

\hypertarget{dcpos}{}\section*{{DCPOs}}\label{dcpos}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definitions}{Definitions}\dotfill \pageref*{definitions} \linebreak
\noindent\hyperlink{the_scott_topology}{The Scott topology}\dotfill \pageref*{the_scott_topology} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A DCPO, or directed-complete partial order, is a [[poset]] with all [[directed joins]]. Often a DCPO is required to have a [[bottom element]] $\bot$; then it is called a pointed DCPO or a CPO (but this term is ambiguous).

The [[morphisms]] between DCPOs preserve the directed joins; equivalently, they are [[Scott continuity|Scott-continuous]]. Morphisms between pointed DCPOs may or may not be required to preserve $\bot$, depending on the application.

In [[domain theory]], a DCPO $P$ is interpreted as a [[type]] (in a programming sense), and its elements are possible partial (in the sense of a [[partial function]]) results of a computation. The bottom element (if there is one) indicates that no result has been obtained; if $x \leq y$ in $P$, then $x$ consists of part of the information in $y$. A [[directed set|directed subset]] $D$ of $P$ indicates a collection of partial results which are mutually consistent, since for any two results $x, y \in D$, there is a partial result that subsumes them both. The required join of $D$ is then a partial result encoding the same information as $D$ itself.

\hypertarget{definitions}{}\subsection*{{Definitions}}\label{definitions}

Recall that a [[poset]] $P$ consists of a collection of [[elements]] equipped with a [[binary relation]] $\leq$ such that $x \leq x$ always and $x \leq z$ whenever $x \leq y$ and $y \leq z$; we also consider $x$ and $y$ to be equal whenever $x \leq y$ and $y \leq x$.

Recall that a [[subset]] $D$ of $P$ is [[directed set|semidirected]] iff, whenever $x, y \in D$, there is some $z \in D$ such that $x \leq z$ and $y \leq z$. Then $D$ is directed iff it is semidirected and [[inhabited subset|inhabited]].

Recall that an [[upper bound]] of a subset $D$ of $P$ is an element $x$ such that $y \leq x$ whenever $y \in D$, and a [[join]] of $D$ is an upper bound $x$ such that $x \leq y$ whenever $y$ is an upper bound of $D$. The join of $D$, if one exists, is unique, and we write it $\bigvee D$ (or even put a little arrow on the right flank of the symbol when $D$ is directed). A [[bottom element]] is a join of the [[empty subset]].

A \textbf{directed-complete partial order} (\_\_DCPO\_\_) or \textbf{predomain} is a poset in which every directed subset has a join. A \textbf{pointed DCPO} or \textbf{complete partial order} (\_\_CPO\_\_) or \textbf{inductive partial order} (\_\_IPO\_\_) or \textbf{semidirected-complete partial order} or \textbf{domain} or \textbf{lift algebra} is a DCPO with a bottom element, equivalently a poset in which every semidirected subset has a join.

\hypertarget{the_scott_topology}{}\subsection*{{The Scott topology}}\label{the_scott_topology}

Every poset $P$ becomes a [[topological space]] under the [[Scott topology]], but this is particularly nice for DCPOs, so we review it.

A [[subset]] $A$ of a DCPO is Scott-[[open subset|open]] iff it's an [[upper subset]] and any directed subset of $P$ whose join belongs to $A$ must meet $A$; it's Scott-[[closed subset|closed]] iff it is a [[lower subset]] that is directed-complete in its own right.

The [[specialisation order]] of the Scott topology of a DCPO is its original order (but the Scott topology is finer than the [[specialisation topology]]).

[[!redirects dcpo]] [[!redirects dcpo's]] [[!redirects dcpos]] [[!redirects DCPO]] [[!redirects DCPO's]] [[!redirects DCPOs]] [[!redirects directed-complete partial order]] [[!redirects directed-complete partial orders]] [[!redirects directed complete partial order]] [[!redirects directed complete partial orders]]

[[!redirects cpo]] [[!redirects cpo's]] [[!redirects cpos]] [[!redirects CPO]] [[!redirects CPO's]] [[!redirects CPOs]] [[!redirects complete partial order]] [[!redirects complete partial orders]]



\end{document}