Įvery order-preserving self-map f of a cpo ( P, ⊥) has a least fixed-point. Thus the complete partial orders with Scott-continuous maps form a cartesian closed category. Equipped with the pointwise order, this is again a dcpo, and a cpo whenever Q is a cpo. The set of all continuous functions between two dcpos P and Q is denoted. This notion of continuity is equivalent to the topological continuity induced by the Scott topology. Note that every continuous function between dcpos is a monotone function.

(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.Ī pointed directed-complete partial order ( pointed dcpo, sometimes abbreviated cppo), is a dcpo with a least element (usually denoted ⊥. The term complete partial order, abbreviated cpo, has several possible meanings depending on context.Ī partially ordered set is a directed-complete partial order ( dcpo) if each of its directed subsets has a supremum. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain theory. 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. It has been suggested that Chain-complete partial order be merged into this article.

Supremo 4.10.3.2151 instaling