Categoricity for primitive recursive and polynomial Boolean algebras

We define a class $\mathbb K_\Sigma$ of primitive recursive structures whose existential diagram is decidable with primitive recursive witnesses. It is proved that a Boolean algebra has a presentation in $\mathbb K_\Sigma$ iff it has a computable presentation with computable set of atoms. Moreover, such a Boolean algebra is primitive recursively categorical with respect to $\mathbb K_\Sigma$ iff it has finitely many atoms. The obtained results can also be carried over to Boolean algebras computable in polynomial time.