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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 ?

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  • $\begingroup$ from your definition, every degree is its own minimal cover. $\endgroup$
    – Denis
    Commented Jan 8, 2014 at 11:08
  • $\begingroup$ You are right. I edited the question. $\endgroup$
    – Archimondain
    Commented Jan 8, 2014 at 11:15
  • $\begingroup$ Then your question does not make sense, because it is now impossible for a degree to have itself as strong minimal cover. $\endgroup$
    – Denis
    Commented Jan 8, 2014 at 11:19
  • $\begingroup$ By "below b", I don't mean "strictly below b" (I assume that is what you disagree with). I'll edit that so there is no confusion. $\endgroup$
    – Archimondain
    Commented Jan 8, 2014 at 11:29
  • $\begingroup$ The problem was not here. In your definition $a$ has to be strictly above $b$, so $a$ cannot be equal to $b$. $\endgroup$
    – Denis
    Commented Jan 8, 2014 at 12:29

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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.

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  • $\begingroup$ For a graphical view of how these classes of Turing degrees fit together one can consult the Computability Menagerie (math.wisc.edu/~jmiller/menagerie.html) $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Jan 8, 2014 at 15:57
  • $\begingroup$ Thanks for the answer and for the link and references. I did not see the question asked in the litterature, I guess it's because it's too soon (in term of the knowledges we have so far on the subject) to be asked. Then there is the question of existence of strong minimal cover for countably many degrees (I think it should be called strong minimal upper bound then), but that is another problem... $\endgroup$
    – Archimondain
    Commented Jan 8, 2014 at 22:07

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