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MODAL OPERATORS ON RINGS OF CONTINUOUS FUNCTIONS

Part of: Ordered structures General logic Maps and general types of spaces defined by maps

Published online by Cambridge University Press:  08 October 2021

GURAM BEZHANISHVILI ,
LUCA CARAI  and
PATRICK J. MORANDI Open the ORCID record for PATRICK J. MORANDI [Opens in a new window]
GURAM BEZHANISHVILI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITYLAS CRUCES, NM 88003, USAE-mail:guram@nmsu.edu
LUCA CARAI*
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITYLAS CRUCES, NM 88003, USA DIPARTIMENTO DI MATEMATICA UNIVERSITÀ DEGLI STUDI DI SALERNO84084FISCIANO (SA), ITALY
PATRICK J. MORANDI
Affiliation:
DEPARTMENT OF MATHEMATICAL SCIENCES NEW MEXICO STATE UNIVERSITYLAS CRUCES, NM 88003, USAE-mail:pmorandi@nmsu.edu
*
E-mail:lcarai@unisa.it
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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 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 ${\sf KHaus}$ of compact Hausdorff spaces and the category $\boldsymbol {\mathit {uba}\ell }$ of uniformly complete bounded archimedean $\ell $ -algebras. We endow a bounded archimedean $\ell $ -algebra with a modal operator, which results in the category $\boldsymbol {\mathit {mba}\ell }$ of modal bounded archimedean $\ell $ -algebras. Our main result establishes a dual adjunction between $\boldsymbol {\mathit {mba}\ell }$ and the category ${\sf KHF}$ 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 ${\sf KHF}$ and the reflective subcategory $\boldsymbol {\mathit {muba}\ell }$ of $\boldsymbol {\mathit {mba}\ell }$ consisting of uniformly complete objects of $\boldsymbol {\mathit {mba}\ell }$ . This generalizes both Gelfand duality and Jónsson-Tarski duality.

Keywords

Modal algebraKripke framereal-valued functionℓ-algebracompact Hausdorff spacecontinuous relation

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 54C30: Real-valued functions 06F25: Ordered rings, algebras, modules 06E25: Boolean algebras with additional operations (diagonalizable algebras, etc.) 06E15: Stone spaces (Boolean spaces) and related structures
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 4 , December 2022 , pp. 1322 - 1348
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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