MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

It is a classic result in modal logic, often referred to as Jonsson-Tarski duality, that the category of modal algebras is dually equivalent to the category of descriptive frames. The latter are Kripke frames equipped with a Stone topology such that the binary relation is continuous. This duality generalizes the celebrated Stone duality for boolean algebras. Our goal is to generalize descriptive frames so that the topology is an arbitrary compact Hausdorff topology. For this, instead of working with the boolean algebra of clopen subsets of a Stone space, we work with the ring of continuous real-valued functions on a compact Hausdorff space. The main novelty is to define a modal operator on such a ring utilizing a continuous relation on a compact Hausdorff space. Our starting point is the well-known Gelfand duality between the category KHaus of compact Hausdorff spaces and the category ubat of uniformly complete bounded archimedean 8-algebras. We endow abounded archimedean 8-algebra with a modal operator, which results in the category mbat of modal bounded archimedean 8-algebras. Our main result establishes a dual adjunction between mbat and the category KHF of what we call compact Hausdorff frames; that is, Kripke frames equipped with a compact Hausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between KHF and the reflective subcategory mubat of mbat consisting of uniformly complete objects of mbat. This generalizes both Gelfand duality and Jonsson-Tarski duality.