Double Stone algebras associated with a topology

Abstract

A double Stone algebra is a Stone algebra whose dual lattice also satisfies the Stone property; this structure was formally introduced by R. Balbes and G. Grätzer (“Injective and projective Stone algebras,” Duke Math. J., vol. 38, pp. 339–347, 1971). The aim of this paper is to develop a construction of the localization of double Stone algebra (or localization of dS-algebra) L of a dS-algebra L, relative to a topology  on L, which represents the first localization of an algebra with two negations. To this end, we introduce the notion of ∧-closed subsets and define a suitable congruence relation that enables the construction of the corresponding algebra of fractions. We also define the concept of -multipliers and the topology associated with a given ∧-closed subset, which additionally omits the paraconsistent elements with respect to ¬. Finally, we prove that the dS-algebra of fractions SC, associated with a ∧-closed subset C of L, is isomorphic to the localization of dS-algebra LC determined by the corresponding topology C.