Topological Dualities for Strong Monadic Distributive Lattices and Applications
Aldo V. Figallo and Inés B. Pascual
In this article, we investigate a subvariety of monadic distributive lattices whose elements we call strong monadic distributive lattices (sM−lattices). Our interest to study them derives from the fact that, to some extent, they are close to monadic Boolean algebras. Indeed, sM-lattices satisfy all the properties that hold in monadic Boolean algebras which do not involve the negation operation. More precisely, sM-lattices are monadic distributive lattices satisfying the identity: ∀(x ∨ ∀y) = ∀x ∨ ∀y. Our main aim is to characterize simple and subdirectly irreducible but not simple sM-lattices. In order to do this, three topological dualities for these algebras are described. Finally, we consider the particular case of finite algebras.
Keywords: Bounded distributive lattices, monadic distributive lattices, Priestley spaces, subdirectly irreducible algebras.