Slender classes

Abstract

A Π10 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 [Trans. Amer. Math. Soc. 353 (2001) 4899–4924] proved that a Π10 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 Π10 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.