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On the structure of the Medvedev lattice

  • Sebastiaan A. Terwijn (a1)
Abstract

We investigate the structure of the Medvedev lattice as a partial order. We prove that every interval in the lattice is either finite, in which case it is isomorphic to a finite Boolean algebra, or contains an antichain of size . the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size . and in fact that these big chains occur in every infinite interval. We also study embeddings of lattices and algebras. We show that large Boolean algebras can be embedded into the Medvedev lattice as upper semilattices, but that a Boolean algebra can be embedded as a lattice only if it is countable. Finally we discuss which of these results hold for the closely related Muchnik lattice.

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[1]Binns, Stephen and Simpson, Stephen G., Embeddings into the Medvedev and Muchnik lattices of classes, Archive for Mathematical Logic, vol. 43 (2004), pp. 399414.
[2]Dyment, Elena Z., Certain properties of the Medvedev lattice, Mathematics of the USSR Sbornik, vol. 30 (1976), pp. 321340, English translation.
[3]Kunen, Kenneth, Set theory: An introduction to independence proofs, North-Holland, 1983.
[4]Medvedev, Yuri T., Degrees of difficulty of the mass problems, Doklady Akademii Nauk SSSR, vol. 104 (1955), no. 4, pp. 501504.
[5]Muchnik, Albert A., Negative answer to the problem of reducibility in the theory of algorithms, Doklady Akademii Nauk SSSR (N.S.), vol. 108 (1956), pp. 194197.
[6]Muchnik, Albert A., On strong and weak reducibility of algorithmic problems, Sibirskii Mathematicheskii Zhurnal. vol. 4 (1963). pp. 13281341, in Russian.
[7]Odifreddi, Piergiorgio G., Classical recursion theory, I, Studies in logic and the foundations of mathematics, vol. 125, North-Holland, 1989.
[8]Odifreddi, Piergiorgio G., Classical recursion theory, II, Studies in logic and the foundations of mathematics, vol. 143, North-Holland, 1999.
[9]Platek, Richard A., A note on the cardinality of the Medvedev lattice. Proceedings of the American Mathematical Society, vol. 25 (1970), p. 917.
[10]Rogers, Hartley Jr., Theory of recursive functions and effective computability, McGraw-Hill, 1967.
[11]Sacks, Gerald E., On suborderings of degrees of recursive unsolvability, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 4656.
[12]Simpson, Stephen, Mass problems and randomness, The Bulletin of Symbolic Logic, vol. 11 (2005), no. 1, pp. 127.
[13]Sorbi, Andrea, Some remarks on the algebraic structure of the Medvedev lattice, this Journal, vol. 55 (1990), no. 2, pp. 831853.
[14]Sorbi, Andrea, The Medvedev lattice of degrees of difficulty, Computability, enumerability, unsolvability: Directions in recursion theory (Cooper, S. B., Slaman, T. A., and Wainer, S. S., editors), London Mathematical Society Lecture Notes, vol. 224, Cambridge University Press, 1996, pp. 289312.
[15]Sorbi, Andrea and Terwijn, Sebastiaan A., Intermediate logics and factors of the Medvedev lattice, to appear in Annals of Pure and Applied Logic.
[16]Terwijn, Sebastiaan A., The finite intervals of the Muchnik lattice, posted on arXiv, 06 28, 2006.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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