Special Splitting in the 2-c.e. Turing Degrees

In this paper, we study a special splitting of the 2-c.e. Turing degrees. The special splitting means a splitting which has a c.e. splitting part. We investigate an existence of a special splitting, including the special splitting with the upper cone avoidance. We prove that there exists a 2-c.e. degree without special splitting and that there is a 2-c.e. degree which has no special splitting avoiding upper cone of some incomputable c.e. degree.