Splitting theorems and the jump operator (Q1295399)

Two sets \(A_1\) and \(A_2\) split \(A\) (written: \(A=A_1\sqcup A_2\)) if \(A=A_1\cup A_2\), and \(A_1\cap A_2=\emptyset\). This splitting is called proper if both \(A_1\) and \(A_2\) are noncomputable. J. B. Remmel asked if a high computably enumerable (c.e.) set can be split into two (c.e.) sets one of which is high. The authors give a negative answer to this question by constructing a high c.e. set all of whose proper splittings consist of \(\text{low}_2\) c.e. sets.