nLab
dcpo

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 $\perp$; 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-continuous?. Morphisms between pointed DCPOs may or may not be required to preserve $\perp$, 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 $x\le y$ in $P$, then $x$ consists of part of the information in $y$. A 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.

Definitions

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

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

Recall that an upper bound of a subset $D$ of $P$ is an element $x$ such that $y\le x$ whenever $x\in D$, and a join of $D$ is an upper bound $x$ such that $x\le y$ 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 directed-complete partial order (__DCPO__) or predomain is a poset in which every directed subset has a join. A pointed DCPO or complete partial order (__CPO__) or inductive partial order (__IPO__) or semidirected-complete partial order or domain or lift algebra is a DCPO with a bottom element, equivalently a poset in which every semidirected subset has a join.

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 iff its an upper subset and any directed subset of $P$ whose join belongs to $A$ must meet $A$; it's Scott-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).