Slender classes

A Π 1 0 class P is called thin if, given a subclass P ′ of P , there is a clopen C with ′ = P ∩ C . Cholak, Coles, Downey and Herrmann [ Trans. Amer. Math. Soc. 353 (2001) 4899–4924] proved that a Π 1 0 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 thin classes are automorphic in the lattice of Π 1 0 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 [ J. London Math. Soc. 70 (2004) 735–749] by showing that this is the only time the Boolean algebra determines the automorphism type of a thin class.