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.
The term complete partial order, abbreviated cpo, has several possible meanings depending on context.
Formulated differently, a pointed dcpo has a supremum for every directed or empty subset.
The term chain-complete partial order is also used, because of the characterization of pointed dcpos as posets in which every chain has a supremum.
A related notion is that of ω-complete partial order (ω-cpo).
[1] Every dcpo is an ω-cpo, since every ω-chain is a directed set, but the converse is not true.
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.
However, this concept occurs far less frequently in practice, since one usually can work on the dual order explicitly.
As a corollary, an ordered set is a pointed dcpo if and only if every (possibly empty) chain has a supremum, i.e., if and only if it is chain-complete.
Thus the complete partial orders with Scott-continuous maps form a cartesian closed category.
This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice.