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Hilbert Algebras with Hilbert–Galois Connections


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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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 ( funded by Horizon 2020 of the European Union.

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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).

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