Scott continuity

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In mathematics, a function between two partially ordered sets P and Q is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D:

$\sup f(D) = f(\sup D).$

Equivalently, a function f between partially ordered sets is Scott-continuous if it is continuous with respect to the Scott topology, a topology on partially ordered sets where a subset O of a partially ordered set is called open if it is an upper set and if all directed sets D with supremum in O have non-empty intersection with O.

Every Scott-continuous function is monotonic.

[edit] Examples

For CPO, the cartesian closed category of complete partial orders, two particularly notable examples of Scott-continuous functions are curry and apply.^[1]

Scott-continuous functions show up in the study of the denotational semantics of computer programs.