On Splits of Computably enumerable sets

TL;DR

This paper investigates various types of splits of computably enumerable sets, providing new proofs and results about the existence and uniformity of such splits, including non-trivial and Friedberg splits.

Contribution

It offers a new proof of Shavrukov's characterization of c.e. sets with non-trivial non-Friedberg splits and shows the impossibility of a uniform splitting for all c.e. sets.

Findings

Every non-computable c.e. set has a non-trivial Friedberg split

A new proof of Shavrukov's result on non-trivial non-Friedberg splits

No uniform splitting exists that satisfies all the specified conditions

Abstract

Our focus will be on the computably enumerable (c.e.) sets and trivial, non-trivial, Friedberg, and non-Friedberg splits of the c.e. sets. Every non-computable set has a non-trivial Friedberg split. Moreover, this theorem is uniform. V. Yu. Shavrukov recently answered the question which c.e. sets have a non-trivial non-Friedberg splitting and we provide a different proof of his result. We end by showing there is no uniform splitting of all c.e. sets such that all non-computable sets are non-trivially split and, in addition, all sets with a non-trivial non-Friedberg split are split accordingly.