Geodesic
Isolated 2-computably enumerable $Q$-degrees
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2010), pp. 3-9

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We demonstrate that for every pair of computably enumerable degrees $\mathbf a$ there exists a properly 2-computably enumerable degree $\mathbf d$, $\mathbf a$, such that $\mathbf a$ isolates $\mathbf d$ from below and $\mathbf b$ isolates $\mathbf d$ from above. As a corollary we prove that there exists a 2-computably enumerable degree which is $Q$-incomparable with any nontrivial (i.e., different from $\boldsymbol0$ and $\boldsymbol0'$) computably enumerable degree, and that every nontrivial computably enumerable degree isolates some 2-computably enumerable degree from below and some 2-computably enumerable degree from above.
Keywords: computably enumerable sets, quasi-reducibility, 2-computably enumerable sets, isolated degrees. 
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     author = {I. I. Batyrshin},
     title = {Isolated 2-computably enumerable $Q$-degrees},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--9},
     publisher = {mathdoc},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2010_4_a0/}
}
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I. I. Batyrshin. Isolated 2-computably enumerable $Q$-degrees. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2010), pp. 3-9. http://geodesic.mathdoc.fr/item/IVM_2010_4_a0/