  The study of the theory of operators over modal pseudocomplemented De Morgan
algebras was begun in the papers [15] and [16]. In this paper, we introduce and
study the class of modal pseudocomplemented De Morgan algebras enriched by an
automorphism k-periodic (or Ck-algebras) where k is a positive integer; for k =
2 the class coincides with the one studied in [15] where the automorphism works
as a new unary operator. In the first place, we prove the class Ck-algebras is
a semisimple variety and we determine the generating algebras. Afterwards, we
calculate the cardinality of the free Ck-algebra with n generator. After the
algebraic study, we built two sentential logics that have as algebraic
counterpart the class of Ck-algebras that we denote L<k and Lk for every k. Lk
is a 1-assertional logic and L<k is the degree-preserving logic both associated
with the class of Ck-algebras. Working over these logics, we prove that L<k is
paraconsistent, which is protoalgebraic and finitely equivalential but not
algebraizable. In contrast, we prove that Lk is algebraizable, sharing the same
theorem with L<k, but not paraconsistent.
