Hyper-MacNeille Completions of Heyting Algebras

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Balbes, R., and Ph. Dwinger, Distributive Lattices, University of Missouri Press, Columbia, Mo., 1974.

    Google Scholar 

  2. 2.

    Belardinelli, F., P. Jipsen, and H. Ono, Algebraic aspects of cut elimination, Studia Logica 77(2):209–240, 2004.

    Article  Google Scholar 

  3. 3.

    Bezhanishvili, G., J. Harding, J. Ilin, and F. M. Lauridsen, MacNeille transferability and stable classes of Heyting algebras, Algebra Universalis 79:55, 2018.

  4. 4.

    Bezhanishvili, N., Lattices of Intermediate and Cylindric Modal Logics, Ph.D. Thesis, University of Amsterdam, 2006.

  5. 5.

    Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, vol. 53 of Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, Cambridge, 2001.

  6. 6.

    Burris, S., and H. P. Sankappanavar, A Course in Universal Algebra, vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York-Berlin, 1981.

  7. 7.

    Burris, S., and H. Werner, Sheaf constructions and their elementary properties, Transactions of the American Mathematical Society 248(2):269–309, 1979.

    Article  Google Scholar 

  8. 8.

    Ciabattoni, A., N. Galatos, and K. Terui, From axioms to analytic rules in nonclassical logics, in Proceedings of the Twenty-Third Annual IEEE Symposium on Logic in Computer Science, LICS 2008, 24-27 June 2008, Pittsburgh, PA, USA, IEEE Computer Society, 2008, pp. 229–240.

  9. 9.

    Ciabattoni, A., N. Galatos, and K. Terui, MacNeille completions of FL-algebras, Algebra Universalis 66(4):405–420, 2011.

    Article  Google Scholar 

  10. 10.

    Ciabattoni, A., N. Galatos, and K. Terui, Algebraic proof theory for substructural logics: cut-elimination and completions, Annals of Pure and Applied Logic 163(3):266–290, 2012.

    Article  Google Scholar 

  11. 11.

    Ciabattoni, A., N. Galatos, and K. Terui, Algebraic proof theory: hypersequents and hypercompletions, Annals of Pure and Applied Logic 168(3): 693–737, 2017.

    Article  Google Scholar 

  12. 12.

    Ciabattoni, A., P. Maffezioli, and L. Spendier, Hypersequent and labelled calculi for intermediate logics, in Automated Reasoning with Analytic Tableaux and Related Methods, vol. 8123 of Lecture Notes in Comput. Sci., Springer, Heidelberg, 2013, pp. 81–96.

  13. 13.

    Comer, S. D., Pacific Journal of Mathematics 38: Representations by algebras of sections over Boolean spaces, 29–38, 1971.

    Article  Google Scholar 

  14. 14.

    Conradie, W., S. Ghilardi, and A. Palmigiano, Unified correspondence, in Johan van Benthem on Logic and Information Dynamics, vol. 5 of Outst. Contrib. Log., Springer, Cham, 2014, pp. 933–975.

  15. 15.

    Conradie, W., and A. Palmigiano, Algorithmic correspondence and canonicity for distributive modal logic, Annals of Pure and Applied Logic 163(3):338–376, 2012.

    Article  Google Scholar 

  16. 16.

    Conradie, W., A. Palmigiano, and S. Sourabh, Algebraic modal correspondence: Sahlqvist and beyond, Journal of Logical and Algebraic Methods in Programming 91:60–84, 2017.

    Article  Google Scholar 

  17. 17.

    Crown, G. D., J. Harding, and M. F. Janowitz, Boolean products of lattices, Order 13(2):175–205, 1996.

    Article  Google Scholar 

  18. 18.

    Davey, B. A., \({\mathfrak{M}}\)-Stone lattices, Canadian Journal of Mathematics 24:1027–1032, 1972.

  19. 19.

    Davey, B. A., Sheaf spaces and sheaves of universal algebras, Mathematische Zeitschrift 134:275–290, 1973.

    Article  Google Scholar 

  20. 20.

    Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, second edn., Cambridge University Press, New York, 2002.

    Book  Google Scholar 

  21. 21.

    Dăneţ, N., The Dedekind completion of \(C(X)\) with pointwise discontinuous functions, in Ordered Structures and Applications, Trends Math., Birkhäuser/Springer, Cham, 2016, pp. 111–126.

    Google Scholar 

  22. 22.

    Esakia, L., Heyting Algebras I. Duality Theory, “Metsniereba”, Tbilisi, 1985. (Russian).

  23. 23.

    Esakia, L., Heyting Algebras: Duality Theory, vol. 50 of Trends in Logic, Springer, 2019. Translated from the Russian by A. Evseev.

  24. 24.

    Galatos, N., and P. Jipsen, Residuated frames with applications to decidability, Transactions of the American Mathematical Society 365(3):1219–1249, 2013.

    Article  Google Scholar 

  25. 25.

    Galatos, N., and H. Ono, Cut elimination and strong separation for substructural logics: an algebraic approach, Annals of Pure and Applied Logic 161(9):1097–1133, 2010.

    Article  Google Scholar 

  26. 26.

    Gehrke, M., The order structure of Stone spaces and the \(T_D\)-separation axiom, Zeitschrift für mathematische Logik und Grundlagen der Mathematik 37(1):5–15, 1991.

    Article  Google Scholar 

  27. 27.

    Gehrke, M., and J. Harding, Bounded lattice expansions, Journal of Algebra 238(1):345–371, 2001.

    Article  Google Scholar 

  28. 28.

    Gehrke, M., J. Harding, and Y. Venema, MacNeille completions and canonical extensions, Transactions of the American Mathematical Society 358(2):573–590, 2006.

    Article  Google Scholar 

  29. 29.

    Gehrke, M., and S. J. van Gool, Sheaves and duality, Journal of Pure and Applied Algebra 222(8):2164–2180, 2018.

    Article  Google Scholar 

  30. 30.

    Givant, S., and Y. Venema, The preservation of Sahlqvist equations in completions of Boolean algebras with operators, Algebra Universalis 41(1):47–84, 1999.

    Article  Google Scholar 

  31. 31.

    Grätzer, G., and E. T. Schmidt, On a problem of M. H. Stone, Acta Mathematica Academiae Scientiarum Hungarica 8:455–460, 1957.

    Article  Google Scholar 

  32. 32.

    Greco, G., P. Jipsen, F. Liang, A. Palmigiano, and A. Tzimoulis, Algebraic proof theory for LE-logics, 2018. ArXiv:1808.04642v1.

  33. 33.

    Hansoul, G., and L. Vrancken-Mawet, Décompositions booléennes de lattis distributifs bornés, Bulletin de la Société Royale des Sciences de Liége 53(2):88–92, 1984.

    Google Scholar 

  34. 34.

    Harding, J., Completions of orthomodular lattices. II, Order 10(3):283–294, 1993.

    Article  Google Scholar 

  35. 35.

    Harding, J., A regular completion for the variety generated by the three-element Heyting algebra, Houston Journal of Mathematics 34(3):649–660, 2008.

    Google Scholar 

  36. 36.

    Harding, J., and G. Bezhanishvili, MacNeille completions of Heyting algebras, Houston Journal of Mathematics 30(4):937–952, 2004.

    Google Scholar 

  37. 37.

    Jónsson, B., Algebras whose congruence lattices are distributive, Mathematica Scandinavica 21:110–121, 1967.

    Google Scholar 

  38. 38.

    Lauridsen, F. M., Cuts and Completions: Algebraic aspects of structural proof theory, Ph.D. Thesis, University of Amsterdam, 2019.

  39. 39.

    Lauridsen, F. M., Intermediate logics admitting a structural hypersequent calculus, Studia Logica 107(2):247–282, 2019.

    Article  Google Scholar 

  40. 40.

    Maehara, S., Lattice-valued representation of the cut-elimination theorem, Tsukuba Journal of Mathematics 15(2):509–521, 1991.

    Article  Google Scholar 

  41. 41.

    Okada, M., Phase semantic cut-elimination and normalization proofs of first- and higher-order linear logic, Theoretical Computer Science 227(1-2):333–396, 1999.

  42. 42.

    Okada, M., A uniform semantic proof for cut-elimination and completeness of various first and higher order logics, Theoretical Computer Science 281(1-2):471–498, 2002.

    Article  Google Scholar 

  43. 43.

    Okada, M., and K. Terui, The finite model property for various fragments of intuitionistic linear logic, Journal of Symbolic Logic 64(2):790–802, 1999.

    Article  Google Scholar 

  44. 44.

    Ono, H., Closure operators and complete embeddings of residuated lattices, Studia Logica 74(3):427–440, 2003.

    Article  Google Scholar 

  45. 45.

    Ono, H., and Y. Komori, Logics without the contraction rule, Journal of Symbolic Logic 50(1):169–201, 1985.

    Article  Google Scholar 

  46. 46.

    Pierce, R. S., Modules over Commutative Regular Rings, Memoirs of the American Mathematical Society, No. 70, Amer. Math. Soc., Providence, R.I., 1967.

  47. 47.

    Priestley, H. A., Ordered topological spaces and the representation of distributive lattices, Proceedings of the London Mathematical Society 24(3):507–530, 1972.

  48. 48.

    Rump, W., and Y. C. Yang, Lateral completion and structure sheaf of an Archimedean \(l\)-group, Journal of Pure and Applied Algebra 213(1):136–143, 2009.

    Article  Google Scholar 

  49. 49.

    Sankappanavar, H. P., Heyting algebras with dual pseudocomplementation, Pacific Journal of Mathematics 117(2):405–415, 1985.

    Article  Google Scholar 

  50. 50.

    Speed, T. P., Some remarks on a class of distributive lattices, Journal of the Australian Mathematical Society 9:289–296, 1969.

    Article  Google Scholar 

  51. 51.

    Speed, T. P., Two congruences on distributive lattices, Bulletin de la Société Royale des Sciences de Liége 38:86–95, 1969.

    Google Scholar 

  52. 52.

    Terui, K., Herbrand’s Theorem via Hypercanonical Extensions, presentation delivered on September 24 at TbiLLC 2013, Gadauri, 2013.

  53. 53.

    Terui, K., MacNeille completion and Buchholz’ omega rule for parameter-free second order logics, in D. Ghica, and A. Jung, (eds.), 27th EACSL Annual Conference on Computer Science Logic (CSL 2018), Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2018, pp. 37:1–37:19.

  54. 54.

    Theunissen, M., and Y. Venema, MacNeille completions of lattice expansions, Algebra Universalis 57(2):143–193, 2007.

    Article  Google Scholar 

  55. 55.

    Vaggione, D. J., Locally Boolean spectra, Algebra Universalis 33(3):319–354, 1995.

    Article  Google Scholar 

  56. 56.

    Wille, A. M., A Gentzen system for involutive residuated lattices, Algebra Universalis 54(4):449–463, 2005.

    Article  Google Scholar 

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to F. M. Lauridsen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Presented by Hiroakira Ono; December 23, 2019.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

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

Mathematics Subject Classification

  • 06D20
  • 06B23
  • 06D15