Finitely generated codings and the degrees r.e. in a degree

We introduce finitely generated (partial) lattices which can be used to code an arbitrary set D. Results of Lerman, Shore and Soare are used to embed these lattices in the degrees r.e. in D. Thus if the degrees r.e. in and above d are isomorphic to those r.e. in and above c, d and c are of the same arithmetic degree. Similar applications are given to generic degrees and general homogeneity questions. After the fact one might say that the main purpose of this paper is to introduce some schemes for doing coding in degree theory via finitely generated sets of degrees as opposed to the usual methods that employ definable substructures. Indeed we will describe such schemes and a number of applications to problems in degree theory. Truthfully, however, the motivation for this paper was the conjecture of Sacks [1966, p. 171] that RED(a), the degrees recursively enumerable in and above a, are isomorphic to RED = RED(0) for every degree a. Our main result will refute this conjecture. Indeed we will show that if RED(a) _ RED(b) then a and b are contained in the same arithmetric degree. As we have said that Sacks [1966] supplies the question behind this paper we should also note that Lerman, Shore and Soare [1981] will essentially supply the answer. In that paper we proved that the r.e. degrees are not Ko-categorical by embedding distinct partial lattices gn in RED, all of which were generated (under A and V) by three elements. We will here use the natural limit P^ of these structures as the core of our coding scheme. Before describing 'r', we restate the definition of a partial lattice. DEFINITION 1. A structure ?' = KP, is a partial lattice if there is a partial order on P containing < and disjoint from $ such that (1.1) Va, b, c E P (a V b = c c is the least upper bound of a and b in P), (1.2) Va, b, c E P (a A b = c c is the greatest lower bound of a and b in P). (Note that we are treating A and V as partial functions.) Now P, has three generators td, to and to. Its elements are ton, tn , t , bgb and b2n for n E N. The defining relations determining the structure of 'r', are as follows. Received by the editors October 30, 1980 and, in revised form, March 10, 1981. Presented to the Society at the Special Session in Recursion Theory at the October 1980 meeting in Kenosha, Wisconsin. 1980 Mathematics Subject Classification. Primary 03D25, 03D30.