Splitting into degrees with low computational strength

Abstract

We investigate the extent to which a c.e. degree can be split into two smaller c.e. degrees which are computationally weak. In contrast to a result of Bickford and Mills that 0′ can be split into two superlow c.e. degrees, we construct a SJT-hard c.e. degree which is not the join of two superlow c.e. degrees. We also prove that every high c.e. degree is the join of two array computable c.e. degrees, and that not every high2 c.e. degree can be split in this way. Finally we extend a result of Downey, Jockusch and Stob by showing that no totally ω-c.a. wtt-degree can be cupped to the complete wtt-degree.