Splitting recursively enumerable subalgebras in recursive Boolean algebras

Abstract

A Boolean algebraB=\(\left\langle {B, \wedge , \vee ,\neg } \right\rangle \) is recursive ifB is a recursive subset of ω and the operations Λ, v and ┌ are partial recursive. A subalgebraC ofB is recursive an (r.e.) ifC is a recursive (r.e.) subset of B. Given an r.e. subalgebraA, we sayA can be split into two r.e. subalgebrasA1 andA2 if (A1 ∪A2)*=A andA1 ∩A2={0, 1}. In this paper we show that any nonrecursive r.e. subalgebra ofB can be split into two nonrecursive r.e. subalgebras ofB. This is a natural analogue of the Friedberg's splitting theorem in ω recursion theory.