Constructivizability of the Boolean algebra $\mathfrak B(\omega)$ with a distinguished automorphism

A constructivizability criterion for the Boolean algebra $\mathfrak B(\omega)$ with a distinguished ideal is given. As a consequence of the criterion, combined with a result due to I. Kalimullin, B. Khoussainov, and A. Melnikov, we construct a Boolean algebra with a distinguished ideal whose degree spectrum contains every nonzero Turing $\Delta^0_2$-degree but does not contain $0$.