Computable Boolean Algebras and Complexity of the Fréchet Ideal

The concept of computability of Boolean algebras is widely used in the theory of constructive and computable models, initiated by A. I. Mal’tsev in the 1960s and actively developed in recent decades. In this paper, we study the long-standing question concerning the decidability of the Fréchet ideal and connections with the decidability of the set of atoms.