  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
2^2^\aleph_0, the size of the lattice itself. We also prove that it is
consistent that the lattice has chains of this size, and in fact that these big
chains occur in every interval that has a big antichain. 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.
