Lattices of c-degrees
Robert S. Lubarsky · Annals of Pure and Applied Logic · 1987
Constructions of minimal Turing degrees have been generalized to embed partial orders as initial segments.Lachlan [4], , Lerman [6,7], and Abraham-Shore [2] have been quite successful in getting positive results, the strongest formulation being that every col-sized locally countable upper semilattice is an ideal in the Turing degrees.Minimal degree proofs carry over almost verbatim to other contexts, such as hyperdegrees, A~-degrees, and c-degrees.In contrast, the differences among these notions express themselves when considering more complicated orders.For instance, Adamowicz [1] uses more complicated machinery to get only that a well-founded upper semi-lattice can be realized via forcing as the structure of the c-degrees.In this paper, we make the distinction among these notions of degree sharper by showing that a general class of partial orders cannot be so realized.Theorem.Suppose U is a countable lattice with a top element, and ~ ~ U. Then U is complete (closed under infinitary ^ and v ).Proof.As a warm-up, we start with some easy cases.Let U = to + to*.Suppose R realizes U. We build a real T which falls in the cut.Identify to with to x to recursively.Let R(0)= R. At stage n, choose the L[R(n)]-least representative of the nth degree and put it in T's nth slot.Let R(n + 1) be the L[R(n)]-least representative of the (n + 1)*th degree.We have coded a representative of each degree from the to-sequence into T, so Vn deg(T)>n.T<~cR(n) Vn since R(n) needs only the finitely many choices of the representatives for the kth degrees, k < n, to replicate the construction.This is a contradiction.Let U-1 + Q + 1.Let R realize U.The former proof won't work directly, since there is no canonical isomorphism between Q and the c-degrees.If we merely choose an (to + to*)-sequence around an irrational cut, there is no * Research supported by NSF grant DMS 84-14103.The author would like to thank Professor Richard Shore for his assistance in the formulation and solution of this problem.