Definitions
A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. Recall that a subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the set. In the literature, dcpos sometimes also appear under the label up-complete poset or simply cpo.
The phrase ω-cpo (or just cpo) is used to describe a poset in which every ω-chain (x1≤x2≤x3≤x4≤...) has a supremum. Every dcpo is an ω-cpo, since every ω-chain is a directed set, but the converse is not true.
An important role is played by dcpo's with a least element. They are sometimes called pointed dcpos, or cppos, or just cpos.
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.
Read more about this topic: Complete Partial Order
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