So it may be anything - for example the program while yes do skip od which goes straight into a tight unending loop. See Exercise 3.21 for a notion of domain based on this fact. Lift Y . We tend to identify y Y with lift y Lift Y . It is important to appreciate that the order implicit here is the extension or information order, not the arithmetical one.
X X be an w -continuous function.
The latter is due to Jan Brouwer, which is ironic because such results rely on excluded middle, and the fixed points do not depend continuously on the given endofunction. Tarski's theorem does assign fixed points continuously - a fact which is crucial to denotational semantics. We shall return to partial functions in Sections 5.3 and 6.3 .