Subsets of hypersimple sets (Q1066890)

It is shown that lattice properties of r.e. sets can affect the degrees of splittings. In particular, if A is r.e. hypersimple then there exists an r.e. set B with \(\emptyset <_ TB<_ TA\) such that if \(A_ 1\coprod A_ 2=A\) is an r.e. splitting of A then \(A_ 1\not\equiv_ TB\). (That is, A does not have the ''universal splitting property''.) It is also shown that this is - in a sense - the best possible result since there do exist (promptly) simple sets with the universal splitting property. This last result combines with one of Maass to show that the universal splitting property is not invariant under automorphisms of the lattice of r.e. sets.