The abbreviation **cpo** stands for “complete partial order”, but confusingly it does *not* generally refer to a partial order that is complete in the sense that the latter word is used in category theory. (A partial order that is complete as a category is instead generally called a complete lattice.)

Instead, a “cpo” generally refers to either a poset that has suprema for directed subsets (a.k.a. a **dcpo**) or for chains indexed by the ordinal $\omega$ (a.k.a. an $\omega$-**cpo** — a weaker condition). Note also that suprema are colimits in a poset rather than limits, so this is actually a cocompleteness condition rather than a completeness one.

The morphisms between cpos are usually taken to be the the Scott-continuous functions (continuous functions with respect to the Scott topology). Cpos and Scott-continuous maps form a category $CPO$.

Cpos are important in domain theory.

- Jiri Adamek, Horst Herrlich, and George Strecker,
*Abstract and concrete categories: the joy of cats*. free online

Last revised on July 11, 2019 at 22:12:14. See the history of this page for a list of all contributions to it.