On the structure of polynomial time degrees

Abstract

The main results of this paper are the following. 1) For both the polynomial time many-one and the polynomial time Turing degrees of recursive sets, every countable distributive lattice can be embedded in any interval of degrees. Furthermore, certain restraints — like preservation of the least or greatest element — can be imposed on the embeddings. 2) The upper semilattice of polynomial time many-one degrees is distributive, whereas that of the polynomial time Turing degrees is nondistributive. This gives the first (elementary) difference between the algebraic structures of p-many-one and p-Turing degrees, respectively.