Towards characterizing the $>\omega^2$-fickle recursively enumerable Turing degrees

TL;DR

This paper explores the structure of recursively enumerable Turing degrees, aiming to identify lattices that characterize degrees above the level of , and investigates the properties of such lattices and upper semilattices.

Contribution

It introduces new lattices related to  levels and examines their embedding properties to characterize higher Turing degrees.

Findings

Identified lattices that characterize  levels.

Discovered three lattices besides M_3 that characterize  levels.

Conjecture that certain upper semilattices have the same degree characterization as their parent lattices.

Abstract

Given a finite lattice $L$ that can be embedded in the recursively enumerable (r.e.) Turing degrees $\mathcal{R}_{\mathrm{T}}$, it is not known how one can characterize the degrees $\mathbf{d}\in\mathcal{R}_{\mathrm{T}}$ below which $L$ can be bounded. Two important characterizations are of the $L_7$ and $M_3$ lattices, where the lattices are bounded below $\mathbf{d}$ if and only if $\mathbf{d}$ contains sets of ``\emph{fickleness}'' $>\omega$ and $\geq\omega^\omega$ respectively. We work towards finding a lattice that characterizes the levels above $\omega^2$, the first non-trivial level after $\omega$. We considered lattices that are as ``short'' and ``narrow'' as $L_7$ and $M_3$, but the lattices characterize also the $>\omega$ or $\geq\omega^\omega$ levels, if the lattices are not already embeddable below all non-zero r.e.\ degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some previously considered lattices, but the removals did not change the levels characterized. This leads us to conjecture that a USL characterizes the same r.e.\ degrees as the lattice it is based on. We discovered three lattices besides $M_3$ that also characterize the $\geq\omega^\omega$-levels. Our search for a $>\omega^2$-candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four $\geq\omega^\omega$-lattices as sublattices.