Abstract
In this paper, we show that the so-called “double bubbles” are not downward dense in the d.c.e. degrees. Here, a pair of d.c.e. degrees \(\mathbf{d}_1> \mathbf{d}_2 > \mathbf{0}\) forms a double bubble if all d.c.e. degrees below \(\mathbf{d}_1\) are comparable with \(\mathbf{d}_2\).
This research was carried out while Yamaleev was visiting the University of Wisconsin under binational NSF grant DMS-1101123 entitled “Collaboration in Computability”. Kuyper’s research was supported by John Templeton Foundation grant 15619: “Mind, Mechanism and Mathematics: Turing Centenary Research Project”. Lempp’s research was partially supported by AMS-Simons Foundation Collaboration Grant 209087. Soskova’s research was supported by Sofia University Science Fund Grant 54/12.04.2016 and by the L’Oréal-UNESCO program “For women in science”. Yamaleev’s research was supported by the Russian Foundation for Basic Research (projects 15-41-02507, 15-01-08252), by the Russian Government Program of Competitive Growth of Kazan Federal University, and by the subsidy allocated to Kazan Federal University for the project part of the state assignment in the sphere of scientific activities (project 1.2045.2014).
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Notes
- 1.
In fact, this idea goes back to Arslanov, who noted it in private communication with Shore. Later, he publicized this idea in conference talks.
References
Arslanov, M.M.: The lattice of the degrees below \({ 0}^{\prime }\), Izv. Vyssh. Uchebn. Zaved. Mat. 7, 27–33 (1988). MR 968729
Arslanov, M.M., Kalimullin, I.S., Lempp, S.: On Downey’s conjecture. J. Symbolic Logic 75(2), 401–441 (2010). MR 2648149
Cooper, S.B.: On a theorem of C. E. M. Yates, handwritten notes
Downey, R.G.: D.r.e. degrees and the nondiamond theorem. Bull. London Math. Soc. 21(1), 43–50 (1989). MR 967789
Ershov, Y.L.: A certain hierarchy of sets. I. Algebra i Logika 7(1), 47–74 (1968). MR 0270911
Ershov, Y.L.: A certain hierarchy of sets. II. Algebra i Logika 7(4), 15–47 (1968). MR 0270912
Ershov, Y.L.: A certain hierarchy of sets. III. Algebra i Logika 9, 34–51 (1970). MR 0299478
Gold, E.M.: Limiting recursion. J. Symbolic Logic 30, 28–48 (1965). MR 0239972
Ishmukhametov, S.T.: On the r.e. predecessors of d.r.e. degrees. Arch. Math. Logic 38(6), 373–386 (1999). MR 1711400
Lachlan, A.H.: Lower bounds for pairs of recursively enumerable degrees. Proc. London Math. Soc. 16(3), 537–569 (1966). MR 0204282
Liu, J., Wu, G., Yamaleev, M.M.: Downward density of exact degrees. Lobachevskii. J. Math. 36(4), 389–398 (2015). MR 3431199
Miller, D.P.: High recursively enumerable degrees, the anti-cup** property, Logic Year 1979–80: University of Connecticut, pp. 230–245 (1981)
Putnam, H.W.: Trial and error predicates and the solution to a problem of Mostowski. J. Symbolic Logic 30, 49–57 (1965). MR 0195725
Sacks, G.E.: On the degrees less than \(0^{\prime }\). Ann. Math. 77(2), 211–231 (1963). MR 0146078
Soare, R.I.: Recursively enumerable sets and degrees: A study of computable functions and computably generated sets. Perspectives in Mathematical Logic, Springer, Berlin (1987). MR 882921
Guohua, W., Yamaleev, M.M.: Isolation: motivations and applications. Uchenye Zapiski Kazanskogo Universiteta 154, 204–217 (2012)
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Andrews, U., Kuyper, R., Lempp, S., Soskova, M.I., Yamaleev, M.M. (2017). Nondensity of Double Bubbles in the D.C.E. Degrees. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_33
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