Autostability relative to strong constructivizations of Boolean algebras with distinguished ideals

We study Boolean algebras with distinguished ideals ($I$-algebras). We proved that a local $I$-algebra is autostable relative to strong constructivizations if and only if it is a direct product of finitely many prime models. We describe complete formulas of elementary theories of local Boolean algebras with distinguished ideals and a finite tuple of distinguished constants. We show that countably categorical $I$-algebras, finitely axiomatizable $I$-algebras, superatomic Boolean algebras with one distinguished ideal, and Boolean algebras are autostable relative to strong constructivizations if and only if they are products of finitely many prime models.