Pseudo-jump inversion and SJT-hard sets

TL;DR

This paper introduces a pseudo-jump operator related to SJT-hard sets, demonstrating its non-invertibility and implications for the structure of computably enumerable sets and their degrees.

Contribution

It defines a new pseudo-jump operator based on SJT-hardness and proves its non-invertibility, revealing limitations in the structure of c.e. degrees.

Findings

The pseudo-jump operator is increasing on all sets.

It cannot be inverted to form a minimal pair.

It cannot avoid an upper cone in the degree structure.

Abstract

There are noncomputable c.e.\ sets, computable from every SJT-hard c.e.\ set. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone.