Extending the Cooper Minimal Pair Theorem

In the study of cappable and noncappable properties of the recursively enumerable (r.e.) degrees, Lempp suggested a conjecture which asserts that for 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 the condition 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 aO and al such that a0, al ≮ b, aO,a1 ≮ h, and aO and a1 form a minimal pair.

每个非零的 a∈ R/ M中不存在极小元(英文)

证明了给定任何非零的递归可枚举图灵度 a存在递归可枚举图灵度 c<a和 d∈M,使得 a≤ d∪ c.由此可以得到 :在每个非零 [a]∈ R∧ M中不存在极小元 ,即给定任何非可盖递归可枚举图灵度 a,存在一个递归可枚举图灵度 c<a,使得 [c]=[a].

Infimum Properties Differ in the Weak Truth-table Degrees and the Turing Degrees

We prove that there are non-recursive r.e.sets A and C with A<T C such that for every set F(?)T A,C∩F≡w(?).