Lattice initial segments of the Turing degrees

TL;DR

This paper characterizes the structure of principal ideals in the Turing degrees below 0', showing they correspond to lattices with a specific computational presentation, and extends the result to degrees above 0".

Contribution

It provides a complete characterization of the isomorphism types of principal ideals in the Turing degrees below 0' as Sigma-0-3 presentable lattices, extending to degrees above 0".

Findings

Principal ideals below 0' are isomorphic to Sigma-0-3 presentable lattices.

The characterization extends to degrees above 0".

Provides a classification framework for Turing degree ideals.

Abstract

We characterize the isomorphism types of principal ideals of the Turing degrees below 0' that are lattices as the lattices with a Sigma-0-3 presentation, by showing that each Sigma-0-3 presentable bounded upper semilattice is isomorphic to such a principal ideal. We get a similar result for the Turing degrees below any degree above 0".