A note on partial numberings

Abstract

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.

An Isomorphism Theorem for Partial Numberings

“…T between sets S and T with numberings ν and κ, respectively, is effective, if there is a function f ∈ P (1) , said to track F , such that ν −1 (dom(F )) ⊆ dom(f ), f (ν −1 (dom(F ))) ⊆ dom(κ), and…”

“…5. ν κ, read ν is computably isomorphic to κ, if there is some one-to-one function g ∈ P (1) with dom(ν) ⊆ dom(g), g(dom(ν)) = dom(κ), and ν n = κ g(n) , for all n ∈ dom(ν).…”

“…The degree structure of partial numberings induced by this reducibility relation has been studied in [1].…”

“…As follows from Definition 2(5), the function g ∈ P (1) witnessing that two partial numberings ν and κ are computably isomorphic is such that for all s ∈ S and all n ∈ dom(g),…”

“…The degree structure of partial numberings is strongly related to that of total numberings in this case. If the weaker reducibility notion is applied, both structures exhibit quite different properties [1].…”