Monadic Distributive Lattices and Monadic Augmented Kripke Frames
A.V. Figallo, I. Pascual and A. Ziliani
In this article, we continue the study of monadic distributive lattices which we began in 1997. These algebras are a natural generalization of monadic Heyting algebras introduced in 1957 by Monteiro and Varsavsky and developed exhaustively by Bezhanishvili between 1998 and 2000. Firstly, we define the category mqP of mq–spaces and mq–functions which is more general than the one considered in 1991 by Cignoli for representing q-distributive lattices. Then, we prove that there is a dual equivalence between mqP and the category mqL of monadic distributive lattices and their corresponding homomorphisms. This duality allows us to characterize in a simple way the subdirectly irreducible algebras in this variety. Next, we introduce the category mAKF whose objects are monadic augmented Kripke frames and whose morphisms are increasing continuous functions verifying certain additional conditions and we prove that mAKF is equivalent to mqP. Finally, we show that the category of perfect augmented Kripke frames, determined by Bezhanishvili in order to represent monadic Heyting algebras, is a proper subcategory of mAKF.
Keywords: Monadic distributive lattices, q−distributive lattices, Priestley spaces, augmented Kripke frames.
AMS subject classification (2010): 03G25; 06D50.