Recursively enumerable sets and degrees
Recursively enumerable sets and degrees
Boolean algebras, Stone spaces, and the iterated Turing jump
Journal of Symbolic Logic
Spectra of highn and non-lown degrees
Journal of Logic and Computation
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Every $\textrm{low}_n$ Boolean algebra, for 1≤n≤4, is isomorphic to a computable Boolean algebra. It is not yet known whether the same is true for n4. However, it is known that there exists a $\textrm{low}_5$ subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, does not have a computable copy. We adapt the proof of this recent result to show that there exists a $\textrm{low}_4$ subalgebra of the computable atomless Boolean algebra which, when viewed as a relation on the computable atomless Boolean algebra, has no computable copy. This result provides a sharp contrast with the one which shows that every $\textrm{low}_4$ Boolean algebra has a computable copy. That is, the spectrum of the subalgebra as a unary relation can contain a $\textrm{low}_4$ degree without containing the degree 0, even though no spectrum of a Boolean algebra (viewed as a structure) can do the same.