# Complete partial order

In mathematics, the phrase **complete partial order** is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain theory.

A **complete partial order** abbreviated **cpo** can, depending on context, refer to any of the following concepts.

Note that *complete partial order* is never used to mean a poset in which *all* subsets have suprema; the terminology complete lattice is used for this concept.

Requiring the existence of directed suprema can be motivated by viewing directed sets as generalized approximation sequences and suprema as *limits* of the respective (approximative) computations. This intuition, in the context of denotational semantics, was the motivation behind the development of domain theory.

The dual notion of a directed-complete poset is called a **filtered-complete partial order**. However, this concept occurs far less frequently in practice, since one usually can work on the dual order explicitly.

An ordered set *P* is a pointed dcpo if and only if every chain has a supremum in *P*, i.e., *P* is chain-complete.^{[5]} Alternatively, an ordered set *P* is a pointed dcpo if and only if every order-preserving self-map of *P* has a least fixpoint.

A function *f* between two dcpos *P* and *Q* is called **(Scott) continuous** if it maps directed sets to directed sets while preserving their suprema:

Note that every continuous function between dcpos is a monotone function. This notion of continuity is equivalent to the topological continuity induced by the Scott topology.

The set of all continuous functions between two dcpos *P* and *Q* is denoted [*P* → *Q*]. Equipped with the pointwise order, this is again a dcpo, and a cpo whenever *Q* is a cpo.
Thus the complete partial orders with Scott-continuous maps form a cartesian closed category.^{[6]}

Every order-preserving self-map *f* of a cpo (*P*, ⊥) has a least fixed-point.^{[7]} If *f* is continuous then this fixed-point is equal to the supremum of the iterates (⊥, *f*(⊥), *f*(*f*(⊥)), … *f*^{n}(⊥), …) of ⊥ (see also the Kleene fixed-point theorem).

Directed completeness alone is quite a basic property that occurs often in other order-theoretic investigations, using for instance algebraic posets and the Scott topology.

Directed completeness relates in various ways to other completeness notions such as chain completeness.