Upper Semilattices in Many-One Degrees

  • Sergei Podzorov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)


The paper gives an overview over recent results of the author on various upper semilattices of many-one degrees. The local isomorphism type (i.e. the collection of isomorphism types of all principal ideals) of m-degrees belonging to any fixed class of arithmetical hierarchy is completely described. The description of the semilattices of simple, hypersimple and \(\Delta^0_2\) m-degrees up to isomorphism is also given.


Distributive Upper Semilattice Many-One Degree Lachlan Semilattice Arithmetical Hierarchy Computably Enumerable Set Simple Set Hypersimple Set Immune Set Hyperimmune Set 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Denisov, S.D.: Structure of the upper semilattice of recursively enumerable m-degrees and related questions. I. Algebra and Logic 17, 418–443 (1978)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Ershov, Y.L.: Hyperhypersimple m-degrees. Algebra i Logika 8, 523–552 [in Russian] (1969) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ershov, Y.L.: The upper semilattice of numerations of a finite set. Algebra and Logic 14, 159–175 (1975)CrossRefGoogle Scholar
  4. 4.
    Ershov, Y.L.: Numbering theory. Nauka, Moskow [in Russian] (1977)Google Scholar
  5. 5.
    Grätzer, G.: Generall lattice theory. Birkhäuser Verlag, Basel (1998)Google Scholar
  6. 6.
    Lachlan, A.: Recursively enumerable many-one degrees. Algebra and Logic 11, 186–202 (1972)CrossRefGoogle Scholar
  7. 7.
    Odifreddi, P.: Classical recursion theory, vol. II. Elsevier, Amsterdam (1999)zbMATHGoogle Scholar
  8. 8.
    Podzorov, S.Y.: On the local structure of Rogers semilattices of Σ 0 n-computable numberings. Algebra and Logic 44, 82–94 (2005)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Podzorov, S.Y.: Numbered distributive semilattices. Siberian Adv. in Math 17, 171–185 (2007)CrossRefGoogle Scholar
  10. 10.
    Podzorov, S.Y.: The universal Lachlan semilattice without the greatest element. Algebra and Logic 46, 163–187 (2007)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Podzorov, S.Y.: Arithmetical m-degrees. Siberian Math. J. (submitted),
  12. 12.
    Rogers, H.: Theory of recursive functions and effective computability. McGraw-Hill Book Company, New York (1967)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Sergei Podzorov
    • 1
  1. 1.Sobolev Institute of MathematicsNovosibirskRussian Federation

Personalised recommendations