Hyper-MacNeille Completions of Heyting Algebras


A Heyting algebra is supplemented if each element a has a dual pseudo-complement \(a^+\), and a Heyting algebra is centrally supplement if it is supplemented and each supplement is central. We show that each Heyting algebra has a centrally supplemented extension in the same variety of Heyting algebras as the original. We use this tool to investigate a new type of completion of Heyting algebras arising in the context of algebraic proof theory, the so-called hyper-MacNeille completion. We show that the hyper-MacNeille completion of a Heyting algebra is the MacNeille completion of its centrally supplemented extension. This provides an algebraic description of the hyper-MacNeille completion of a Heyting algebra, allows development of further properties of the hyper-MacNeille completion, and provides new examples of varieties of Heyting algebras that are closed under hyper-MacNeille completions. In particular, connections between the centrally supplemented extension and Boolean products allow us to show that any finitely generated variety of Heyting algebras is closed under hyper-MacNeille completions.

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The authors would like to thank the anonymous referee for reading the paper carefully and for supplying them with many helpful comments. The second named author would also like to thank Sam van Gool for helpful discussion and for drawing attention to [50, 51]. This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 689176.

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Correspondence to F. M. Lauridsen.

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Presented by Hiroakira Ono; December 23, 2019.

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Harding, J., Lauridsen, F.M. Hyper-MacNeille Completions of Heyting Algebras. Stud Logica (2021). https://doi.org/10.1007/s11225-021-09941-6

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  • Heyting algebra
  • completions
  • MacNeille completion
  • Boolean product
  • sheaf
  • supplemented lattice

Mathematics Subject Classification

  • 06D20
  • 06B23
  • 06D15