Modal operators on rings of continuous functions

TL;DR

This paper generalizes the duality between modal algebras and descriptive frames to a setting involving rings of continuous functions on compact Hausdorff spaces, introducing modal operators via continuous relations.

Contribution

It extends duality theory to compact Hausdorff spaces by defining modal operators on rings of continuous functions, unifying Gelfand duality and modal algebra duality.

Findings

Established a dual adjunction between modal bounded archimedean $ ext{l}$-algebras and compact Hausdorff frames.

Proved a dual equivalence between uniformly complete modal $ ext{l}$-algebras and reflective subcategory of frames.

Generalized classical dualities to a broader topological and algebraic context.

Abstract

It is a classic result in modal logic 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. Our goal is to further 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 $ubal$ of uniformly complete bounded archimedean $\ell$-algebras. We endow a bounded archimedean $\ell$-algebra with a modal operator, which results in the category $mbal$ of modal bounded archimedean $\ell$-algebras. Our main result establishes a dual adjunction between $mbal$ and the category $KHK$ 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 $KHK$ and the reflective subcategory $mubal$ of $mbal$ consisting of uniformly complete objects of $mbal$. This generalizes both Gelfand duality and the duality for modal algebras.