Monadic Boolean algebras with an automorphism and their relation to \({\mathbf{Df}}_{\mathbf{2}}\)-algebras

In this work, we initiate an investigation of the class \({\mathcal {B}}_{T_km}\) of monadic Boolean algebras endowed with a monadic automorphism of period k. These algebras constitute a generalization of monadic symmetric Boolean algebras. We determine the congruences on these algebras and we characterize the subdirectly irreducible algebras. This last result allows us to prove that \({\mathcal {B}}_{T_km}\) is a discriminator variety and as a consequence, the principal congruences are characterized. Finally, we explore, in the finite case, the relationship between this class and the class \({\mathbf{Df}}_{\mathbf{2}}\) of diagonal-free two-dimensional cylindric algebras.