It is a classic result in modal logic, often referred to as Jónsson-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 compactHausdorff 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 thewell-known Gelfand duality between the category KHaus of compactHausdorff spaces and the category ubaℓ of uniformly complete bounded archimedean ℓ-algebras.We endowa bounded archimedean ℓ-algebra with a modal operator, which results in the category mbaℓ of modal bounded archimedean ℓ-algebras. Our main result establishes a dual adjunction between mbaℓ and the category KHF of what we call compactHausdorff frames; that is,Kripke frames equipped with a compactHausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between KHF and the reflective subcategory mubaℓ of mbaℓ consisting of uniformly complete objects of mbaℓ. This generalizes both Gelfand duality and Jónsson-Tarski duality.

Modal operators on rings of continuous functions / G. Bezhanishvili, L. Carai, P.J. Morandi. - In: THE JOURNAL OF SYMBOLIC LOGIC. - ISSN 0022-4812. - 87:4(2022), pp. 1322-1348. [10.1017/jsl.2021.83]

Modal operators on rings of continuous functions

L. Carai;
2022

Abstract

It is a classic result in modal logic, often referred to as Jónsson-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 compactHausdorff 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 thewell-known Gelfand duality between the category KHaus of compactHausdorff spaces and the category ubaℓ of uniformly complete bounded archimedean ℓ-algebras.We endowa bounded archimedean ℓ-algebra with a modal operator, which results in the category mbaℓ of modal bounded archimedean ℓ-algebras. Our main result establishes a dual adjunction between mbaℓ and the category KHF of what we call compactHausdorff frames; that is,Kripke frames equipped with a compactHausdorff topology such that the binary relation is continuous. This dual adjunction restricts to a dual equivalence between KHF and the reflective subcategory mubaℓ of mbaℓ consisting of uniformly complete objects of mbaℓ. This generalizes both Gelfand duality and Jónsson-Tarski duality.
compact Hausdorff space; continuous relation; Kripke frame; Modal algebra; real-valued function; ℓ-algebra
Settore MAT/01 - Logica Matematica
Settore MAT/02 - Algebra
Settore MAT/03 - Geometria
2022
8-ott-2021
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/1018315
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