Cupping and noncupping in the enumeration degrees of ∑20 sets

Cupping and noncupping in the enumeration

We prove the following three theorems on the enumeration degrees of Σ2 sets. Theorem A: There exists a nonzero noncuppable Σ 0 2 enumeration degree. Theorem B: Every nonzero ∆2 enumeration degree is cuppable to 0e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low ∆2 enumeration degree with the anticupping property.

Cupping Classes of Enumeration Degrees

Cupping Classes of Σ0 2 Enumeration Degrees

We prove that no subclass of the Σ 2 enumeration degrees containing the 3-c.e. enumeration degrees can be cupped to 0e by a single Σ 2 enumeration degree.

Cupping D20 Enumeration Degrees to 0 e '

In this paper we prove that every nonzero ∆2 e-degree is cuppable to 0e by a 1-generic ∆ 0 2 e-degree (so low and nontotal) and that every nonzero ω-c.e. e-degree is cuppable to 0e by an incomplete

The limitations of cupping in the local structure of the enumeration degrees

We prove that a sequence of sets containing representatives of cupping partners for every nonzero ∆2 enumeration degree cannot have a ∆ 0 2 enumeration. We also prove that no subclass of the Σ 2 enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to 0e by a single incomplete Σ 2 enumeration degree.