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 Algebra Logika: Year: Volume: Issue: Page: Find

 Algebra Logika, 2002, Volume 41, Number 2, Pages 143–154 (Mi al177)

Friedberg Numberings of Families of $n$-Computably Enumerable Sets

S. S. Goncharova, S. Lemppb, R. Solomonb

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We establish a number of results on numberings, in particular, on Friedberg numberings, of families of d.c.e. sets. First, it is proved that there exists a Friedberg numbering of the family of all d.c.e. sets. We also show that this result, patterned on Friedberg's famous theorem for the family of all c.e. sets, holds for the family of all $n$-c.e. sets for any $n>2$. Second, it is stated that there exists an infinite family of d. c. e. sets without a Friedberg numbering. Third, it is shown that there exists an infinite family of c. e. sets (treated as a family of d. c. e. sets) with a numbering which is unique up to equivalence. Fourth, it is proved that there exists a family of d. c. e. sets with a least numbering (under reducibility) which is Friedberg but is not the only numbering (modulo reducibility).

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English version:
Algebra and Logic, 2002, 41:2, 81–86

Bibliographic databases:

UDC: 510.10+510.57

Citation: S. S. Goncharov, S. Lempp, R. Solomon, “Friedberg Numberings of Families of $n$-Computably Enumerable Sets”, Algebra Logika, 41:2 (2002), 143–154; Algebra and Logic, 41:2 (2002), 81–86

Citation in format AMSBIB
\Bibitem{GonLemSol02} \by S.~S.~Goncharov, S.~Lempp, R.~Solomon \paper Friedberg Numberings of Families of $n$-Computably Enumerable Sets \jour Algebra Logika \yr 2002 \vol 41 \issue 2 \pages 143--154 \mathnet{http://mi.mathnet.ru/al177} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1922986} \zmath{https://zbmath.org/?q=an:1063.03028} \transl \jour Algebra and Logic \yr 2002 \vol 41 \issue 2 \pages 81--86 \crossref{https://doi.org/10.1023/A:1015352513117} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-33846184874} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Badaev S.A., Talasbaeva Zh.T., “Computable numberings in the hierarchy of Ershov”, Mathematical Logic in Asia, 2006, 17–30
2. Brodhead P., Cenzer D., “Effectively closed sets and enumerations”, Archive For Mathematical Logic, 46:7–8 (2008), 565–582
3. S. S. Goncharov, N. T. Kogabaev, “O $\Sigma^0_1$-klassifikatsii otnoshenii na vychislimykh strukturakh”, Vestn. NGU. Ser. matem., mekh., inform., 8:4 (2008), 23–32
4. S. S. Ospichev, “Some Properties of Numberings in Various Levels in Ershov's Hierarchy”, J. Math. Sci., 188:4 (2013), 441–448
5. S. S. Ospichev, “Infinite family of $\Sigma_a^{-1}$-Sets with only One Computable Numbering”, J. Math. Sci., 188:4 (2013), 449–451
6. N. A. Bazhenov, “The branching theorem and computable categoricity in the Ershov hierarchy”, Algebra and Logic, 54:2 (2015), 91–104
7. S. S. Ospichev, “Computable families of sets in Ershov hierarchy without principal numberings”, J. Math. Sci., 215:4 (2016), 529–536
8. S. S. Ospichev, “Friedberg numberings in the Ershov hierarchy”, Algebra and Logic, 54:4 (2015), 283–295
9. Badaev S.A., Manat M., Sorbi A., “Friedberg Numberings in the Ershov Hierarchy”, Arch. Math. Log., 54:1-2 (2015), 59–73
10. Lange K. Miller R. Steiner R.M., “Classifications of Computable Structures”, Notre Dame J. Form. Log., 59:1 (2018), 35–59
11. I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Faizrahmanov, “Positive presentations of families relative to $e$-oracles”, Siberian Math. J., 59:4 (2018), 648–656
12. I. Sh. Kalimullin, V. G. Puzarenko, M. Kh. Faizrakhmanov, “Positive presentations of families in relation to reducibility with respect to enumerability”, Algebra and Logic, 57:4 (2018), 320–323
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