Topological Duality for Nelson Algebras and Its Applications (abstract)

some appropriate axioms (for details see [7]). These axioms imply that the relation ≈ on A defined by: a ≈ b if and only if a → b = 1 and b → a = 1, 1)/ ≈ is a Heyting algebra. For a given Heyting algebra B there always exists a Nelson algebra A such that A h is isomorphic to B: the Fidel-Vakarelov construction of the Nelson algebra N (B) (see e.g. [8]) yields an example of such an algebra. In this paper we describe all Nelson algebras A whose A h 's are isomor-phic to a given Heyting algebra B; and next we consider some problems, related with this description, concerning equational and quasiequational subclasses (= subvarieties and subquasivarieties) of the class N of all Nel-son algebras. Our description is obtained by an application of the topo-logical duality theory of Priestley for bounded distributive lattices ([3], [4], [5]). A partially ordered topological space x is said to be totally order disconnected if for all x, y ∈ X with x ≤ y there exists a clopen increasing set U such that x ∈ U and y ∈ U. X is said to be a h-space if it is a compact totally order disconnected space such that for every open subset U of X, (U ] (= the smallest decreasing subset of X containing U) is open. A continuous order-preserving map f between h-spaces X and Y is defined