Partial Numberings and Precompleteness

Precompleteness is a powerful property of numberings. Most numberings commonly used in computability theory such as the Godel numberings of the partial computable functions are precomplete. As is well known, exactly the precomplete numberings have the effective fixed point property. In this paper extensions of precompleteness to partial numberings are discussed. As is shown, most of the important properties shared by precomplete numberings carry over to the partial case.