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 ) <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant="script">F</m:mi> </m:mrow> </m:msub> </m:math> ${L}_{\mathcal{F}}$ of a dS -algebra L , relative to a topology <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> $\mathcal{F}$ 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> $\mathcal{F}$ -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 S C , associated with a ∧-closed subset C of L , is isomorphic to the localization of dS -algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi mathvariant="script">F</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msub> </m:mrow> </m:msub> </m:math> ${L}_{{\mathcal{F}}_{C}}$ determined by the corresponding topology <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi mathvariant="script">F</m:mi> </m:mrow> <m:mrow> <m:mi>C</m:mi> </m:mrow> </m:msub> </m:math> ${\mathcal{F}}_{C}$ .