Monadic distributive lattices and monadic augmented Kripke frames

TL;DR

This paper extends duality theory for monadic distributive lattices, characterizes their subdirectly irreducible and finite algebras, and introduces a category of monadic augmented Kripke frames equivalent to the algebraic structures.

Contribution

It generalizes duality results to monadic distributive lattices and introduces a new categorical framework of monadic augmented Kripke frames.

Findings

Extended duality for m-lattices

Characterized subdirectly irreducible and finite m-lattices

Established equivalence between algebraic and Kripke frame categories

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.