Modal operators on rings of continuous functions: A generalization of Gelfand and Jonsson-Tarski dualities

Jonsson-Tarski duality in modal logic generalizes the celebrated Stone duality for Boolean algebras by modeling modal operators on Boolean algebras by means of continuous relations on Stone spaces. But the notion of continuous relation makes perfect sense for the class of compact Hausdorff spaces. Gelfand duality generalizes Stone duality to compact Hausdroff spaces by working with continuous functions instead of clopens. I will discuss how to define modal operators on rings of continuous functions by means of continuous relations on compact Hausdorff spaces. The resulting duality will on the one hand generalize Gelfand duality, and on the other Jonsson-Tarski duality.