Monadic distributive lattices and monadic augmented Kripke frames

Abstract

In this article, we continue the study of monadic distributive lattices (or m-lattices) which are a natural generalization of monadic Heyting algebras, introduced by Monteiro and Varsavsky and developed exhaustively by Bezhanishvili. First, we extended the duality obtained by Cignoli for Q−distributive lattices to m-lattices. This new duality allows us to describe in a simple way the subdirectly irreducible algebras in this variety and in particular, to characterize the finite ones. Next, we introduce the category mKF whose objects are monadic augmented Kripke frames and whose morphisms are increasing continuous functions verifying certain additional conditions and we prove that it is equivalent to the one obtained above. Finally, we show that the category of perfect augmented Kripke frames given by Bezhanishvili for monadic Heyting algebras is a proper subcategory of mKF .