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Algebra Logika, 2003, Volume 42, Number 2, Pages 211–226 (Mi al26)  

This article is cited in 7 scientific papers (total in 7 papers)

Initial Segments in Rogers Semilattices of $\Sigma^0_n$-Computable Numberings

S. Yu. Podzorov

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

Abstract: S. Goncharov and S. Badaev showed that for $n\geqslant 2$, there exist infinite families whose Rogers semilattices contain ideals without minimal elements. In this connection, the question was posed as to whether there are examples of families that lack this property. We answer this question in the negative. It is proved that independently of a family chosen, the class of semilattices that are principal ideals of the Rogers semilattice of that family is rather wide: it includes both a factor lattice of the lattice of recursively enumerable sets modulo finite sets and a family of initial segments in the semilattice of $m$-degrees generated by immune sets.

Keywords: Rogers semilattice, recursively enumerable set, immune set, $m$-degree.

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English version:
Algebra and Logic, 2003, 42:2, 121–129

Bibliographic databases:

UDC: 510.5
Received: 19.03.2001

Citation: S. Yu. Podzorov, “Initial Segments in Rogers Semilattices of $\Sigma^0_n$-Computable Numberings”, Algebra Logika, 42:2 (2003), 211–226; Algebra and Logic, 42:2 (2003), 121–129

Citation in format AMSBIB
\Bibitem{Pod03}
\by S.~Yu.~Podzorov
\paper Initial Segments in Rogers Semilattices of $\Sigma^0_n$-Computable Numberings
\jour Algebra Logika
\yr 2003
\vol 42
\issue 2
\pages 211--226
\mathnet{http://mi.mathnet.ru/al26}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2003630}
\zmath{https://zbmath.org/?q=an:1029.03033}
\transl
\jour Algebra and Logic
\yr 2003
\vol 42
\issue 2
\pages 121--129
\crossref{https://doi.org/10.1023/A:1023354407888}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-42349108044}


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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. S. Yu. Podzorov, “Local Structure of Rogers Semilattices of $\Sigma^0_n$-Computable Numberings”, Algebra and Logic, 44:1 (2005), 82–94  mathnet  crossref  mathscinet  zmath
    2. S. Yu. Podzorov, “Arithmetical $D$-degrees”, Siberian Math. J., 49:6 (2008), 1109–1123  mathnet  crossref  mathscinet  isi
    3. N. A. Baklanova, “Nerazreshimost elementarnykh teorii polureshetok Rodzhersa na predelnykh urovnyakh arifmeticheskoi ierarkhii”, Vestn. NGU. Ser. matem., mekh., inform., 11:4 (2011), 3–7  mathnet
    4. S. A. Badaev, S. S. Goncharov, “Generalized computable universal numberings”, Algebra and Logic, 53:5 (2014), 355–364  mathnet  crossref  mathscinet  isi
    5. M. V. Dorzhieva, “Nerazreshimost elementarnykh teorii polureshetok Rodzhersa analiticheskoi ierarkhii”, Sib. elektron. matem. izv., 13 (2016), 148–153  mathnet  crossref
    6. M. Kh. Faizrakhmanov, “Universal computable enumerations of finite classes of families of total functions”, Russian Math. (Iz. VUZ), 60:12 (2016), 79–83  mathnet  crossref  isi
    7. S. A. Badaev, A. A. Issakhov, “Some absolute properties of $A$-computable numberings”, Algebra and Logic, 57:4 (2018), 275–288  mathnet  crossref  crossref  isi
  • Алгебра и логика Algebra and Logic
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