On a conjecture of Lempp

In this paper, we first prove that there exist computably enumerable (c.e.) degrees a and b such that a ≰ b, and for any c.e. degree u, if u ≤ a and u is cappable, then u ≤ b, so refuting a conjecture of Lempp (in Slaman [1996]); secondly, we prove that: (A. Li and D. Wang) there is no uniform construction to build nonzero cappable degree below a nonzero c.e. degree, that is, there is no computable function f such that for all e ∈ ω, (i) W_f(e) ≤_T W_e, (ii) W_f(e) has a cappable degree, and (iii) W_f(e) ≰_T ∅ unless W_e ≰_T ∅.