Strong minimal covers and a question of Yates: The story so far
Andrew E. M. Lewis · Cambridge University Press eBooks · 2009
. An old question of Yates as to whether all minimal degrees have a strong minimal cover remains one of the longstanding problems of degree theory, apparently largely impervious to present techniques. We survey existing results in this area, focussing especially on some recent progress. Introduction . By the 60's and 70's degree theorists had become concerned with some particular and fundamental questions of a global nature concerning the structure of the Turing degrees. In order to address issues regarding homogeneity and the decidability and degree of the theory, the approach taken at this time was to proceed through a deep analysis of the initial segments of the structure. Along these lines a technique for piecemeal construction of initial segments, even if only locally successful, would have been very useful and it was in this context that interest was first aroused in a question of Yates: definition 1.1. A degree b is a strong minimal cover for a if D [< b ] = D [≤ a ]. A degree a is minimal if it is a strong minimal cover for 0. Question 1.1 (Yates). Does every minimal degree have a strong minimal cover? In fact, the question of characterizing those degrees with strong minimal cover had already been raised by Spector in his 1956 paper [CS]. Certainly in D m —the structure of the many-one degrees, induced by a strengthening of the Turing reducibility—Lachlan's proof of the fact that every m -degree has a strong minimal cover played a vital role in Ershov's [YE] and Paliutin's [EP] results characterizing the structure and in showing, for instance, that 0 m is the only definable singleton.