1-Generic splittings of computably enumerable degrees

Say a set G ⊆ ω is 1-generic if for any e ∈ ω , there is a string σ ⊂ G such that { e } σ ( e ) ↓ or ∀ τ ⊇ σ ( { e } τ ( e ) ↑ ) . It is known that 0 ′ can be split into two 1-generic degrees. In this paper, we generalize this and prove that any nonzero computably enumerable degree can be split into two 1-generic degrees. As a corollary, no two computably enumerable degrees bound the same class of 1-generic degrees.