Computable one-to-one enumerations of effective domains

Abstract

In this paper ω-algebraic complete partial orders are considered the compact elements of which are not maximal in the partial order. Under the assumption that these elements are indexed such that their equality is decidable and the order is semi-decidable (completely enumerable) it is shown that the computable domain elements can be effectively enumerated without repetition. Computable one-to-one enumerations of the computable domain elements are minimal among all enumerations of these elements with respect to the reducibility of one enumeration to another. In computability studies of continuous complete partial orders one usually uses a generalization of Gödel numberings, called admissible numberings. They are maximal among the computable enumerations. As it is shown, each admissible numbering is recursively isomorphic to the directed sum of a computable family of computable one-to-one enumerations. Both results generalize well-known theorems of Friedberg and Schinzel respectively for the partial recursive functions. Their premise is satisfied by each type in the hierarchy of the Eršov-Scott higher type partial computable functionals, which means that any such type can be computably enumerated without repetition and any of its admissible indexings is recursively isomorphic to the directed sum of a computable family of computable one-to-one enumerations. The proofs use a priority argument.