Monadic pseudocomplemented distributive lattices

In this paper, we study the variety of pseudocomplemented distributive lattices with existential and universal quantifiers, called monadic pseudocomplemented distributive lattices. We introduce the variety of monadic KAN-algebras, which turns out to be different from the class studied in [Gomez C., Marcos M., San Martín H.J.: \textit{On the relation of negations in Nelson algebras}. Rep. Math. Logic \textbf{56} (2021), 15--56], and prove that the category of monadic pseudocomplemented distributive lattices is equivalent to the category of centered monadic KAN-algebras, extending the results given in [Calomino I., Pelaitay G.: \textit{A new categorical equivalence for Stone algebras}. Accepted in Mathematica Slovaca (2025)].