Journal of Symbolic Logic

On the structure of the Medvedev lattice

Sebastiaan A. Terwijn

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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 220, the size of the lattice itself. We also prove that it is consistent with ZFC that the lattice has chains of size 220, 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.

Article information

Source
J. Symbolic Logic Volume 73, Issue 2 (2008), 543-558.

Dates
First available: 16 April 2008

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1208359059

Digital Object Identifier
doi:10.2178/jsl/1208359059

Mathematical Reviews number (MathSciNet)
MR2414464

Zentralblatt MATH identifier
1140.03020

Subjects
Primary: 03D28: Other Turing degree structures 03D30: Other degrees and reducibilities 03G10: Lattices and related structures [See also 06Bxx]

Keywords
Medvedev degrees chains antichains lattice embeddings

Citation

Terwijn, Sebastiaan A. On the structure of the Medvedev lattice. Journal of Symbolic Logic 73 (2008), no. 2, 543--558. doi:10.2178/jsl/1208359059. http://projecteuclid.org/euclid.jsl/1208359059.


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