The Lattice of Pi-0-1 Classes

			Douglas Cenzer
		    University of Florida

Pi-0-1 classes are important for applications to many areas of
mathematics and are also a central part of computability theory. We
will discuss recent results and open problems in the lattice L of
Pi-0-1 classes. It is interesting to compare and contrast with results
in the lattice of computably enumerable degrees. Issues include
embeddings, automorphisms, invariance, definability, and connections
with Turing degrees.  For example, Cholak, Coles, Downey and Herrmann
showed that perfect thin classes are definable in L and that any two
perfect thin Pi-0-1 classes are automorphic. Let L(P) be the lattice
of Pi-0-1 subclasses of P. Cenzer and Nies showed that any finite
lattice with the (dual) reduction property is isomorphic, modulo
finite difference, to some initial segment L(P) and that for any
decidable Pi-0-1 class, if L(P) is not a Boolean algebra, then the
theory of L(P) is undecidable.