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 refer to any of the following concepts depending on context. A partially ordered set is a directed-complete partial order (dcpo) if each of its directed subsets has a supremum. A subset of a partial order is directed if it is non-empty and every pair of elements has an upper bound in the subset. In the literature, dcpos sometimes also appear under the label up-complete poset. A partially ordered set is a pointed directed-complete partial order if it is a dcpo with a least element. They are sometimes abbreviated cppos. A partially ordered set is a ω-complete partial order (ω-cpo) if it is a poset in which every ω-chain (x1 ≤ x2 ≤ x3 ≤ x4 ≤ ...) has a supremum that belongs to the poset. Every dcpo is an ω-cpo, since every ω-chain is a directed set, but the converse is not true. However, every ω-cpo with a basis is also a dcpo (with the same basis). An ω-cpo (dcpo) with a basis is also called a continuous ω-cpo (continuous dcpo). 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 partial order 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. Every finite poset is directed complete.

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