Degrees of splittings and bases of recursively enumerable subspaces

Abstract

This paper analyzes the interrelationships between the (Turing) of r.e. bases and of r.e. splittings of r.e. vector spaces together with the relationship of the degrees of bases and the degrees of the vector spaces they generate. For an r.e. subspace V V of V ∞ {V_\infty } , we show that α \alpha is the degree of an r.e. basis of V V iff α \alpha is the degree of an r.e. summand of V V iff α \alpha is the degree and dependence degree of an r.e. summand of V V . This result naturally leads to explore several questions regarding the degree theoretic properties of pairs of summands and the ways in which bases may arise.