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Hilbert Algebras with Hilbert–Galois Connections

Studia Logica volume 111, pages 113–138 (2023)Cite this article

Abstract

In this paper we introduce Hilbert algebras with Hilbert–Galois connections (HilGC-algebras) and we study the Hilbert–Galois connections defined in Heyting algebras, called HGC-algebras. We assign a categorical duality between the category HilGC-algebras with Hilbert homomorphisms that commutes with Hilbert–Galois connections and Hilbert spaces with certain binary relations and whose morphisms are special functional relations. We also prove a categorical duality between the category of Heyting Galois algebras with Heyting homomorphisms that commutes with Hilbert–Galois connections and the category of spectral Heyting spaces endowed with a binary relation with certain special continuous maps.

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Acknowledgements

We would like to thank the referees for the comments and suggestions on the presentation of this paper. The authors acknowledge partial support of the project PICT 2019-00882 (2021-2023) CaToAM: triple abordaje semántico de las lógicas modales multivaluadas. The second author was supported by the MSCA-RISE - Marie Skłodowska-Curie Research and Innovation Staff Exchange (RISE) Project MOSAIC No. 101007627 (https://doi.org/10.3030/101007627) funded by Horizon 2020 of the European Union.

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Authors and Affiliations

  1. Dpto. de Matemática, Facultad de Economía y Administración and Instituto de Investigación en Tecnología y Ciencias de la Ingeniería (IITCI), Universidad Nacional del Comahue and CONICET, Neuquén, Argentina

    Daniela Montangie

  2. Dpto. de Matemática, Facultad de Ciencias Exactas, CONICET and Universidad Nacional del Centro de la Provincia de Buenos Aires, Tandil, Argentina

    Sergio A. Celani

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  1. Sergio A. Celani
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  2. Daniela Montangie
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Correspondence to Daniela Montangie.

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Presented by Yde Venema; Received October 15, 2020.

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Celani, S.A., Montangie, D. Hilbert Algebras with Hilbert–Galois Connections. Stud Logica 111, 113–138 (2023). https://doi.org/10.1007/s11225-022-10019-0

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Keywords

  • Hilbert algebras
  • Hilbert–Galois connections
  • Topological representation
  • Heyting algebras