### On the structure of the Medvedev lattice

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

#### 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.

First Page:
Primary Subjects: 03D28, 03D30, 03G10
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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

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