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

Monadic distributive lattices and monadic augmented Kripke frames