Hyperarithmetical Categoricity of the Boolean Algebra $\mathfrak{B}(\omega^{\alpha}\times\eta)$

We consider $\Delta^{0}_{\beta}$-categoricity in Boolean algebras. We prove the following theorem: if $\delta$ is a limit ordinal or 0, $n\in\omega$, and $\delta+n\geqslant 1$, then the Boolean algebra $\mathfrak{B}(\omega^{\delta+n}\times\eta)$ is $\Delta^{0}_{\delta+2n+1}$-categorical and not $\Delta^{0}_{\delta+2n}$-categorical.