Partial numberings

What is it about?

The different behaviour of total and partial numberings with respect to the reducibility preorder is investigated. Partial numberings appear quite naturally in computability studies for topological spaces. The degrees of partial numberings form a distributive lattice which in the case of an infinite numbered set is neither complete nor contains a least element. Friedberg numberings are no longer minimal in this situation. Indeed, there is an infinite descending chain of non-equivalent Friedberg numberings below every given numbering, as well as an uncountable antichain.

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Why is it important?

Partial numberings show up in recent research on continuous data like the computable real numbers. Classical numbering theory has been developed as a tool for computable algebra and has mainly considered total numberings. As is pointed out in the note, partial numberings behave differently from total ones in many aspects and need further study.