References
S. S. Goncharov and A. Sorbi, “Generalized computable numerations and nontrivial Rogers semilattices,” Algebra and Logic, 36, No. 6, 359-369 (1997).
Yu. L. Ershov, The Theory of Numberings [in Russian], Nauka, Moscow (1977).
S. A. Badaev and S. S. Goncharov, “Theory of numberings: Open problems,” in Computability Theory and Its Applications, Cont. Math., 257, Am. Math. Soc., Providence, RI (2000), pp. 23-38.
S. A. Badaev and S. S. Goncharov, “Computability and numberings,” in New Computational Paradigms, S. B. Cooper, B. Lowe, and A. Sorbi (eds.), Springer, New York (2008), pp. 19-34.
J. C. Owings, Jr., “The meta-r.e. sets, but not the \( {\Pi}_1^1 \) sets, can be enumerated without repetition,” J. Symb. Log., 35, No. 2, 223-229 (1970).
S. A. Badaev, S. S. Goncharov, and A. Sorbi, “Isomorphism types of Rogers semilattices for families from different levels of the arithmetical hierarchy,” Algebra and Logic, 45, No. 6, 361-370 (2006).
S. Yu. Podzorov, “Arithmetical D-degrees,” Sib. Math. J., 49, No. 6, 1109-1123 (2008).
I. Herbert, S. Jain, S. Lempp, M. Mustafa, and F. Stephan, “Reductions between types of numberings,” Ann. Pure Appl. Log., 170, No. 12 (2019), Article 102716, pp. 1-25.
N. Bazhenov, S. Ospichev, and M. Yamaleev, “Isomorphism types of Rogers semilattices in the analytical hierarchy,” 2019, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singapore, accepted; arXiv:1912.05226 [math.LO].
S. A. Badaev, S. S. Goncharov, S. Yu. Podzorov, and A. Sorbi, “Algebraic properties of Rogers semilattices of arithmetical numberings,” in Computability and Models, S. B. Cooper and S. S. Goncharov (eds.), Kluwer Academic/Plenum Publishers, New York (2003), pp. 45-77.
M. V. Dorzhieva, “Undecidability of elementary theory of Rogers semilattices in analytical hierarchy,” Sib. El. Mat. Izv., 13, 148-153 (2016); http://semr.math.nsc.ru/v13/p148-153.pdf.
N. Bazhenov and M. Mustafa, “Elementary theories of Rogers semilattices in the analytical hierarchy,” submitted.
N. Bazhenov and M. Mustafa, “Rogers semilattices in the analytical hierarchy: The case of finite families,” submitted.
S. A. Badaev, S. S. Goncharov, and A. Sorbi, “Isomorphism types and theories of Rogers semilattices of arithmetical numberings,” in Computability and Models, S. B. Cooper and S. S. Goncharov (eds.), Kluwer Academic/Plenum Publishers, New York (2003), pp. 79-91.
T. Jech, Set Theory, Springer Monogr. Math., Springer, Berlin (2003).
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Translated from Algebra i Logika, Vol. 59, No. 5, pp. 594-599, September-October, 2020. Russian DOI: https://doi.org/10.33048/alglog.2020.59.506.
N. A. Bazhenov is The study was carried out within the framework of the state assignment to Sobolev Institute of Mathematics SB RAS, project No. 0314-2019-0002.
M. Mustafa is Supported by Nazarbayev University FDCRGP, N090118FD5342.
S. S. Ospichev is Supported by RFBR, project No. 20-01-00300.
M. M. Yamaleev is Supported by a Funding Program for the Regional Scientific and Educational Mathematical Center of the Volga Federal Region, Agreement No. 075-02-2020-1478.
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Bazhenov, N.A., Mustafa, M., Ospichev, S.S. et al. Numberings in the Analytical Hierarchy. Algebra Logic 59, 404–407 (2020). https://doi.org/10.1007/s10469-020-09613-9
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DOI: https://doi.org/10.1007/s10469-020-09613-9