Monadic Distributive Lattices

The purpose of this paper is to investigate the variety of algebras, which we call monadic distributive lattices, as a natural generalization of monadic Heyting algebras [16]. It is worth mentioning that the latter is a proper subvariety of the first one, as it is shown in a simple example. Our main interest is the characterization of simple and subdirectly irreducible monadic distributive lattices. In order to do this, a duality theory for these algebras is developed. The duality enables us to describe the lattice of congruences on monadic distributive lattices. Finally, our attention is focused upon the relationship between the category of dual spaces associatted with these algebras and the category of perfect Ono frames considered by Bezhanishvili in order to represent monadic Heyting algebras.