Towards Finding a Lattice That Characterizes the >ω²-Fickle Recursively Enumerable Turing Degrees

Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees (R_T, ≤_T), we do not in general know how to characterize the degrees d ε R_T below which L can be bounded. The important characterizations known are of the L₇ and M₃ lattices, where the lattices are bounded below d if and only if d contains sets of 'fickleness' >ω and ≥ω^ω respectively. We work towards finding a lattice that characterizes the levels above ω² , the first non-trivial level after ω. We introduced a lattice-theoretic property called '3-directness' to describe lattices that are no 'wider' or 'taller' than L₇ and M₃ . We exhaust the 3-direct lattices L, but they turn out to also characterize the >ω or ≥ω^ω levels, if L is not already embeddable below all non-zero r.e. degrees. We also considered upper semilattices (USLs) by removing the bottom meet(s) of some 3-direct 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 on which the USL is based. We discovered three 3-direct lattices besides M₃ that also characterize the ≥ω^ω -levels. Our search for a >ω²-candidate therefore involves the lattice-theoretic problem of finding lattices that do not contain any of the four ≥ω^ω-lattices as sublattices.