On the structure of the Medvedev lattice

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.

• Type: Journal
• Year: 2008
• Where: JSYML
• Authors: Sebastiaan Terwijn
• Keywords: Boolean Algebra, Finite Boolean Algebra, JSYML 2008, Medvedev Lattice