Directed complete partial orders

Abbreviation: DCPO

Definition

A directed complete partial order is a poset P = (P, ≤ ) such that every directed subset of P has a least upper bound: ∀D ⊆ P (D ≠ Ø  and  ∀x,y ∈ D ∃z ∈ D (x,y ≤ z)  ⇒  ∃z ∈ P(z = ∨D)).

Morphisms

Let P and Q be directed complete partial orders. A morphism from P to Q is a function f : P→Q that is Scott-continuous, which means that f preserves all directed joins: z = ∨D  ⇒  f(z) = ∨f[D] for all directed sets D ⊆ P.

Some results

Examples

(R, ≤ ), the real numbers with the standard order.

(P(S), ⊆ ), the collection of subsets of a sets S, ordered by inclusion.

Properties

Finite members

Subclasses

Superclasses