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

<p>Given a finite lattice L that can be embedded in the recursively enumerable (r.e.) Turing degrees <i>(<i>R</i>_T, ≤_T)</i>, we do not in general know how to characterize the degrees <b>d</b> <i><b>ε</b></i> <i><i>R</i>_T</i>below which<em> </em><i>L</i> can be bounded. The important characterizations known are of the <i>L₇</i> and <i></i><i>M</i><i>₃</i> 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 <i>L₇</i> and <i>M₃</i> . We exhaust the 3-direct lattices <i>L</i>, but they turn out to also characterize the >ω or ≥ω^ω levels, if <i>L</i> 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 <i>M₃</i> 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.</p>