New results on hierarchies of domains

Abstract

The relation between a cpo D and its space [D → D] of Scott-continuous functions is investigated. For a cpo D with a least element we show that if [D → D] is algebraic then D itself is algebraic. Together with a generalization of Smyth's Theorem to strict functions this implies that [D → D] is ω-algebraic whenever (Formula presented.) is. It is an open question, whether [D → D] is ω-algebraic whenever (Formula presented.) is algebraic. Smyth's Theorem is extended to the class of cpo's with no least element assumed. Our main result asserts that in this context the profinites again form the largest cartesian closed category of domains. The proof also yields the following: if the functionspace of a cpo D is algebraic, then D has infima for filtered sets. The question, whether an ω-algebraic functionspace implies that D is profinite, remains open.