The largest Cartesian closed category of domains, considered constructively

A conjecture of Smyth is discussed which says that if D and [D → D] are effectively algebraic directed-complete partial orders with least element (cpo’s), then D is an effectively strongly algebraic cpo, where it was not made precise what is meant by an effectively algebraic and an effectively strongly algebraic cpo. Notions of an effectively strongly algebraic cpo and an effective SFP domain are introduced and shown to be (effectively) equivalent. Moreover, the conjecture is shown to hold if instead of being effectively algebraic, [D → D] is only required to be ω-algebraic and D is forced to have a completeness test, that is a procedure which decides for any two finite sets X and Y of compact cpo elements whether X is a complete set of upper bounds of Y . As a consequence, the category of effective SFP objects and continuous maps turns out to be the largest Cartesian closed full subcategory of the category of ω-algebraic cpo’s that have a completeness test. It is then studied whether such a result also holds in a constructive framework, where one considers categories with constructive domains as objects, that is, domains consisting only of the constructive (computable) elements of an indexed ω-algebraic cpo, and computable maps as morphisms. This is indeed the case: the category of constructive SFP domains is the largest constructively Cartesian closed weakly indexed effectively full subcategory of the category of constructive domains that have a completeness test and satisfy a further effectivity requirement.