We show that for any computably enumerable (c.e.) degree a and any low n–c.e. degree l (n ≥ 1), if l < a, then there are n–c.e. degrees a0,a1 such that l < a0,a1 < a and a0 ∨ a1 = a. In particular, there is no low maximal d.c.e. degree.

We show that there is a strong minimal pair in the computably enumerable Turing degrees, i.e., a pair of nonzero c.e. degrees a and b such that a ∩ b = 0 and for any nonzero c.e. degree x ≤ a, b ∪ x ≥ a.

The K-trivial sets form an ideal in the Turing degrees, which is generated by its computably enumerable (c.e.) members and has an exact pair below the degree of the halting problem. The question of whether it has an exact pair in the c.e. degrees was first raised in [MN06, Question 4.2] and later in [Nie09, Problem 5.5.8]. We give a negative answer to this question. In fact, we show the followi...