A generalization of de Vries duality to closed relations between compact Hausdorff spaces

Marco Abbadini; Guram Bezhanishvili; Luca Carai

doi:10.1016/j.topol.2023.108641

A generalization of de Vries duality to closed relations between compact Hausdorff spaces

Marco Abbadini, Guram Bezhanishvili, Luca Carai^*

^*Corresponding author for this work

Computer Science

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Abstract

Stone duality generalizes to an equivalence between the categories Stone^R of Stone spaces and closed relations and BA^S of boolean algebras and subordination relations. Splitting equivalences in Stone^R yields a category that is equivalent to the category KHaus^R of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BA^S yields a category that is equivalent to the category De V^S of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHaus^R is equivalent to De V^S, thus resolving a problem recently raised in the literature.

The equivalence between KHaus^R and De V^S further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory De V^F of De V^S whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition.

Original language	English
Article number	108641
Number of pages	22
Journal	Topology and its Applications
Volume	337
Early online date	13 Jul 2023
DOIs	https://doi.org/10.1016/j.topol.2023.108641
Publication status	Published - 1 Sept 2023

Bibliographical note

Acknowledgments:
Marco Abbadini and Luca Carai were supported by the Italian Ministry of University and Research through the PRIN project n. 20173WKCM5 Theory and applications of resource sensitive logics. Luca Carai acknowledges partial support from the Juan de la Cierva-Formación 2021 programme (FJC2021-046977-I) funded by MCIN/AEI/10.13039/501100011033 and by the European Union “NextGenerationEU”/PRTR.

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abstract = "Stone duality generalizes to an equivalence between the categories StoneR of Stone spaces and closed relations and BAS of boolean algebras and subordination relations. Splitting equivalences in StoneR yields a category that is equivalent to the category KHausR of compact Hausdorff spaces and closed relations. Similarly, splitting equivalences in BAS yields a category that is equivalent to the category De VS of de Vries algebras and compatible subordination relations. Applying the machinery of allegories then yields that KHausR is equivalent to De VS, thus resolving a problem recently raised in the literature.The equivalence between KHausR and De VS further restricts to an equivalence between the category KHausR of compact Hausdorff spaces and continuous functions and the wide subcategory De VF of De VS whose morphisms satisfy additional conditions. This yields an alternative to de Vries duality. One advantage of this approach is that composition of morphisms is usual relation composition.",

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A generalization of de Vries duality to closed relations between compact Hausdorff spaces

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Bibliographical note

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