Extending the Cooper Minimal Pair Theorem

In the study of cappable and noncappable properties of the recursivelyenumerable (r.e.) degrees, Lempp suggested a conjecture which asserts thatfor all r.e. degrees a and b, if a!=b then there exists an r.e. degree c such that c<=a and c!=b and c is cappable. We shall prove in this paper that this conjecture holds under thecondition that a is high. Working below a high r.e. degree h,we show that for any r.e. degree b with h!=b,there exist r.e. degrees a0 and a1 such thata0,a1 !=b, a0,a1<=h, a0 and a1 form a minimal pair.