Initial Segments of the Lattice of 0 1 Classes

The study of the lattice E of computably enumerable sets under inclusion has been one of the central tasks of computability theory since the 1960's. We investigate initial segments of the lattice L of 0 1 classes (of sets) under inclusion and we compare this lattice with E. It was recently proved by Nies that the theory of each interval of the lattice E which is not a boolean algebra interprets true arithmetic (and is therefore certainly undecidable.) However, we will show that in L there are initial segments ;; P] = L(P) which are not Boolean algebras, but which have a decidable theory. In fact, we will construct for any nite distributive lattice L which satisses the dual of the usual reduction property a 0 1 class P such that L is isomorphic to the lattice L(P) modulo nite diierences. The construction of the 0 1 class corresponding to a given lattice builds on the construction of a minimal 0 1 class by Cenzer, Downey, Jockusch and Shore in 1993. The simplest minimal 0 1 class P has a single limit point together with countably many isolated points. P has the property that every 0 1 subclass Q of P is either nite or is coonite in P { furthermore, Q is the intersection of P with a clopen set. Thus the lattice 0; P] of 0 1 subclasses of P is isomorphic to the class of nite/coonite subsets of ! and is a Boolean algebra. Such a class plays a role in the lattice L corresponding to the dual of the role played by a maximal c. e. set in the lattice E. 1 For the simplest new 0 1 class P constructed, P has a single, non-computable limit point and L(P) has three elements, corresponding to ;, P and a minimal class P 0 P. The third type of subclass has no complement in the lattice, which is why the lattice is not a Boolean algebra. On the other hand, the theory of L is shown to be decidable, so that this lattice may not be realized as the class of c. e. subsets of any c. e. set. A 0 1 class P is said to be decidable if it is the set of paths through a computable tree T with no dead ends. We show that if P is decidable and has only nitely many …