On the lattice of Pi(0,1) classes

Effectively closed sets, as modeled by $P01$ classes, have played an important role in computability theory going back to the Kleene basis theorem [1955]. Many of the fundamental results about $P01$ classes and their members were established by Jockusch and Soare in [1972]. Cenzer et al. [1999] is a short course on $P01$ classes. The $P01$ classes occur naturally in the application of computability to many areas of mathematics. Cenzer and Remmel [1998] is a recent survey with many examples. Minimal and thin $P01$ classes were investigated by Cenzer et al. [1993]. This dissertation is a comparative study of the lattice $LP$ of $P01$ classes with the lattice $E$ of computably enumerable (c.e.) sets. The work in this thesis concerns the lattice of $P01$ classes (modulo finite difference), compared and contrasted with the lattice of c.e. sets. The notion of a minimal extension Q of a class P is defined to mean that there is no class strictly between P and Q. Previously only trivial examples were known, but here we give general conditions under which P has a minimal extension. Recently initial segments of the lattice (that is, subsets of a given set) have been studied. It was shown, in contrast to the lattice of c.e. sets, that a finite lattice can be realized which is not a Boolean algebra; in particular, any finite ordinal can be realized. This thesis announces an improvement of these results by constructing a $P01$ class P such that the family of subclasses of P is isomorphic to the smallest infinite ordinal (ω). Also studied are definability of various properties (such as finiteness) and invariance under automorphism.