Slender Classes . Rod Downey And

A Π1 class P is called thin if, given a subclass P ′ of P there is a clopen C with P ′ = P ∩C. Cholak, Coles, Downey and Herrmann [7] proved that a Π1 class P is thin if and only if its lattice of subclasses forms a Boolean algebra. Those authors also proved that if this boolean algebra is the free Boolean algebra, then all such think classes are automorphic in the lattice of Π1 classes under inclusion. From this it follows that if the boolean algebra has a finite number n of atoms then the resulting classes are all automorphic. We prove a conjecture of Cholak and Downey [8] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class.