Continuous Functions and Fixpoints
A function f between two dcpos P and Q is called (Scott) continuous if it maps directed sets to directed sets while preserving their suprema:
- is directed for every directed .
- for every directed .
Note that every continuous function between dcpos is a monotone function. This notion of continuity is equivalent to the topological continuity induced by the Scott topology.
The set of all continuous functions between two dcpos P and Q is denoted . Equipped with the pointwise order, this is again a dcpo, and a cpo whenever Q is a cpo. Thus the complete partial orders with Scott continuous maps form a cartesian closed category.
Every order-preserving self-map f of a cpo (P, ⊥) has a least fixpoint. If f is continuous then this fixpoint is equal to the supremum of the iterates (⊥, f(⊥), f(f(⊥)), … fn(⊥), …) of ⊥ (see also the Kleene fixpoint theorem).
Read more about this topic: Complete Partial Order
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