Randomness and the linear degrees of computability

We show that there exists a real a such that, for all reals beta, if alpha is linear reducible to beta (alpha <= l beta, previously denoted as alpha <=(sw) beta) then beta <=(T) alpha. In fact, every random real satisfies this quasi-maximality property. As a corollary we may conclude that there exists no e-complete Delta(2) real. Upon realizing that quasi-maximality does not characterize the random reals - there exist reals which are not random but which are of quasi-maximal e-degree - it is then natural to ask whether maximality could provide such a characterization. Such hopes, however, are in vain since no real is of maximal e-degree. (c) 2006 Elsevier B.V. All rights reserved.