Low_n Boolean Subalgebras

摘要

Every 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 n > 4. However, it is known that there exists a lows 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 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 low_4 Boolean algebra has a computable copy. That is, the spectrum of the subalgebra as a unary relation can contain a low_4 degree without containing the degree 0, even though no spectrum of a Boolean algebra (viewed as a structure) can do the same.