Splitting and nonsplitting in the Σ20 enumeration degrees

How enumeration reducibility yields extended Harrington non-splitting

Sacks [14] showed that every computably enumerable (c.e.) degree > 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. sp...

Bounding nonsplitting enumeration degrees

Total Degrees and Nonsplitting Properties of Enumeration Degrees

The Strongest Nonsplitting Theorem

Sacks [14] showed that every computably enumerable (c.e.) degree ≥ 0 has a c.e. splitting. Hence, relativising, every c.e. degree has a Δ2 splitting above each proper predecessor (by ‘splitting’ we understand ‘nontrivial splitting’). Arslanov [1] showed that 0′ has a d.c.e. splitting above each c.e. a < 0′. On the other hand, Lachlan [9] proved the existence of a c.e. a > 0 which has no c.e. sp...

A non-splitting theorem in the enumeration degrees

We complete a study of the splitting/non-splitting properties of the enumeration degrees below 0′e by proving an analog of Harrington’s non-splitting theorem for the Σ2 enumeration degrees. We show how non-splitting techniques known from the study of the c.e. Turing degrees can be adapted to the enumeration degrees.