We say that a Turing degree $a$ is a strong minimal cover of $b$ if $a$ is strictly above $b$ and if any $c$ strictly below $a$ is (not necessarily strictly) below $b$. It is known that some degrees do have strong minimal covers and some other don't.


It is known that 0 has strong minimal covers. Those strong minimal covers are called minimal degrees. It is (as far as I know) an open question to know whether every minimal degree has itself a strong minimal cover.


My question is : can the question about minimal degree be extended to every degree which is a strong minimal cover of some other degree ? or is it known (for obvious or complicated reasons) that there is some strong minimal covers which don't themselves have strong minimal cover ?



Is there a Turing degree which is a strong minimal cover and does not have itself a strong minimal cover?
Not known, I think. Traditional ways to produce a SMC $ a $ by completely controlling the structure of $[0, a] $ (starting with Spector 1956, see Lerman's 1983 book) led to c.e. traceable degrees, but those all have SMCs as shown by Ishmukhametov, 1999. Nontraditional way would be to follow Kumabe's construction of a minimal degree that is DNC (diagonally non-computable) (which implies not c.e. traceable as shown by Stephan in a joint paper with me and Merkle) but recent work of Cai et al. indicates that such constructions also tend to give degrees that have SMCs. Lewis-Pye has many related results. 


