A partially ordered set is a directed complete partial order (dcpo) if each of its directed subsets has a supremum. A complete partial order (cpo) is a dcpo with a least element. In the literature, dcpos sometimes also appear under the label up-complete poset or, confusingly, simply "cpo". A dcpo with a least element is sometimes called a pointed dcpo or a pointed cpo.
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 cpo if and only if every chain has a supremum in P. Alternatively, an ordered set P is a cpo if and only if every order-preserving self-map of P has a least fixpoint. Every set S can be turned into a cpo by adding a least element ⊥ and introducing a flat order with ⊥ ≤ s for every s ∈ S and no other order relations.
A function f between two dcpos P and Q is called continuous if it maps directed sets to directed sets while preserving their suprema:
A function f between two cpos (P, ⊥P) and (Q, ⊥Q) is called continuous if it furthermore preserves the least element:
This notion of continuity is equivalent to that 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 a dcpo, and a cpo whenever Q is a cpo.
Every continuous function between cpos is order-preserving but not vice versa. Every order-preserving self-map f of a cpo (P, ⊥) has a least fixpoint. If f is continuous then this fixpoint is equal to the supremum of the iterates (⊥, f(⊥), f(f(⊥)), … fn(⊥), …) of ⊥ (see also the Kleene fixpoint theorem).
Directed completeness relates in various ways to other completeness notions. 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. Functions between dcpos are often required to be monotone or even Scott continuous. The complete partial orders with Scott continuous maps form a cartesian closed category, denoted CPO.