On the triple jump of the set of atoms of a Boolean algebra

Abstract

We prove the following result concerning the degree spectrum of the atom relation on a computable Boolean algebra. Let $\mathcal {C}$ be a computable Boolean algebra with infinitely many atoms and $\mathbf {a}$ be the Turing degree of the atom relation of $\mathcal {C}$. If $\mathbf {d}$ is a c.e. degree such that $\mathbf {a}^{\prime \prime \prime }\leq _T\mathbf {d}^{\prime \prime \prime }$, then there is a computable copy of $\mathcal {C}$ where the atom relation has degree $\mathbf {d}$. In particular, for every $\mathrm {high}_3$ c.e. degree $\mathbf {d}$, any computable Boolean algebra with infinitely many atoms has a computable copy where the atom relation has degree $\mathbf {d}$.