HKSS-completeness of modal algebras

The paper studies computability-theoretic properties of countable modal algebras. We prove that the class of modal algebras is complete in the sense of the work of Hirschfeldt, Khoussainov, Shore, and Slinko. This answers an open question of Bazhenov [Stud. Log., 104 (2016), 1083–1097]. The result implies that every degree spectrum and every categoricity spectrum can be realized by a suitable modal algebra.