A DCPO, or directed-complete partial order, is a poset with all directed joins. Often a DCPO is required to have a bottom element ; 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 , depending on the application.
In domain theory, a DCPO 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 in , then consists of part of the information in . A directed subset of indicates a collection of partial results which are mutually consistent, since for any two results , there is a partial result that subsumes them both. The required join of is then a partial result encoding the same information as itself.
Recall that an upper bound of a subset of is an element such that whenever , and a join of is an upper bound such that whenever is an upper bound of . The join of , if one exists, is unique, and we write it (or even put a little arrow on the right flank of the symbol when 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.
A subset of a DCPO is Scott-open iff its an upper subset and any directed subset of whose join belongs to must meet ; it's Scott-closed iff it is a lower subset that is directed-complete in its own right.