Paraconsistent and Paracomplete Logics Based on k-Cyclic Modal Pseudocomplemented De Morgan Algebras

The study of the theory of operators over modal pseudocomplemented De Morgan algebras was begun in papers [20] and [21]. In this paper, we introduce and study the class of modal pseudocomplemented De Morgan algebras enriched by a k-periodic automorphism (or C-k-algebras). We denote by inverted left perpendicular(k) the automorphism where k is a positive integer. For k=2, the class coincides with the one studied in [20] where the automorphism works as a new unary operator which can be considered as a negation. In the first place, we develop an algebraic study of the class of C-k-algebras; as consequence, we prove the class C-k-algebras is a semisimple variety and determine the generating algebras. After doing the algebraic study and using these properties, we built two families of sentential logics that we denote with L-k(<=) and L-k for every k. L-k is a 1-assertional logic and L-k(<=) is the degree-preserving logic both associated with the class of C-k-algebras. Working over these logics, we prove that L-k(<=) is paraconsistent with respect to the de Morgan negation similar to, which is protoalgebraic and finitely equivalential but not algebraizable. In contrast, we prove that L-k is algebraizable, sharing the same theorems with L-k(<=) , but not paraconsistent with respect to similar to. Furthermore, we show that L-k(<=) and L-k are paracomplete logics with respect to similar to and inverted left perpendicular(k) and paraconsistent logics with respecto to inverted left perpendicular(k), for every k.