Skip to main content
SpringerLink
Log in

Canonical extensions and ultraproducts of polarities

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

Jónsson and Tarski’s notion of the perfect extension of a Boolean algebra with operators has evolved into an extensive theory of canonical extensions of lattice-based algebras. After reviewing this evolution we make two contributions. First it is shown that the failure of a variety of algebras to be closed under canonical extensions is witnessed by a particular one of its free algebras. The size of the set of generators of this algebra can be made a function of a collection of varieties and is a kind of Hanf number for canonical closure. Secondly we study the complete lattice of stable subsets of a polarity structure, and show that if a class of polarities is closed under ultraproducts, then its stable set lattices generate a variety that is closed under canonical extensions. This generalises an earlier result of the author about generation of canonically closed varieties of Boolean algebras with operators, which was in turn an abstraction of the result that a first-order definable class of Kripke frames determines a modal logic that is valid in its so-called canonical frames.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almeida, A.: Canonical extensions and relational representations of lattices with negation. Stud. Log. 91, 171–199 (2009)

    MathSciNet  MATH  Google Scholar 

  2. Bayart, A.: Quasi-adéquation de la logique modal du second ordre S5 et adéquation de la logique modal du premier ordre S5. Logique et Analyse 2(6–7), 99–121 (1959)

    Google Scholar 

  3. Bell, J.L., Slomson, A.B.: Models and Ultraproducts. North-Holland, Amsterdam (1969)

    MATH  Google Scholar 

  4. Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions and canonical extensions of Heyting algebras. Order 23, 143–161 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Bezhanishvili, G., Mines, R., Morandi, P.J.: Topo-canonical completions of closure algebras and Heyting algebras. Algebra Universalis 58, 1–34 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Birkhoff, G.: Lattice Theory, 1st edn. American Mathematical Society, New York (1940)

  7. Bruns, G., Roddy, M.: A finitely generated modular ortholattice. Can. Math. Bull. 35(1), 29–33 (1992)

    MathSciNet  MATH  Google Scholar 

  8. Bulian, J., Hodkinson, I.: Bare canonicity of representable cylindric and polyadic algebras. Anna. Pure Appl. Logic 164(9), 884–906 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer-Verlag, New York (1981)

    MATH  Google Scholar 

  10. Burris, S., Werner, H.: Sheaf constructions and their elementary properties. Trans. Am. Math. Soc. 248, 269–309 (1979)

    MathSciNet  MATH  Google Scholar 

  11. Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  12. Chernilovskaya, A., Gehrke, M., van Rooijen, L.: Generalized Kripke semantics for the Lambek-Grishin calculus. Logic J. IGPL 20(6), 1110–1132 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Conradie, W., Frittella, S., Palmigiano, A., Piazzai, M., Tzimoulis, A., Wijnberg, N.M.: Categories: How I learned to stop worrying and love two sorts. In: Väänänen, J. (ed.) WoLLIC 2016. Lecture Notes in Computer Science, vol. 9803, pp. 145–164. Springer-Verlag, New York (2016)

    Google Scholar 

  14. Conradie, W., Palmigiano, A.: Algorithmic correspondence and canonicity for non-distributive logics. arXiv:1603.08515 (2016)

  15. Coumans, D.: Generalising canonical extension to the categorical setting. Ann. Pure Appl. Logic 163(12), 1940–1961 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Coumans, D., Gehrke, M., van Rooijen, L.: Relational semantics for full linear logic. J. Appl. Logic 12(1), 50–66 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Craig, A.: Canonical extensions of bounded lattices and natural duality for default bilattices. Ph.D. thesis, University of Oxford (2012)

  18. Craig, A., Haviar, M.: Reconciliation of approaches to the construction of canonical extensions of bounded lattices. Math. Slovaca 64(6), 1335–1356 (2014)

    MathSciNet  MATH  Google Scholar 

  19. Craig, A.P.K., Haviar, M., Priestley, H.A.: A fresh perspective on canonical extensions for bounded lattices. Appl. Categorical Struct. 21, 725–749 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Cresswell, M.J.: A Henkin completeness theorem for T. Notre Dame J. Formal Logic 8, 186–190 (1967)

    MathSciNet  MATH  Google Scholar 

  21. Davey, B.A., Haviar, M., Priestley, H.A.: Boolean topological distributive lattices and canonical extensions. Appl. Categorical Struct. 15, 225–241 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Davey, B.A., Priestley, H.: A topological approach to canonical extensions in finitely generated varieties of lattice-based algebras. Topol. Appl. 158, 1724–1731 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Davey, B.A., Priestley, H.: Canonical extensions and discrete dualities for finitely generated varieties of lattice-based algebras. Studia Logica 100, 137–161 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  25. Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symbol. Logic 70(3), 713–740 (2005)

    MathSciNet  MATH  Google Scholar 

  26. Fine, K.: Logics containing K4. Part I. J. Symbol. Logic 39(1), 31–42 (1974)

    MathSciNet  MATH  Google Scholar 

  27. Fine, K.: Some connections between elementary and modal logic. In: S. Kanger (ed.) Proceedings of the Third Scandinavian Logic Symposium, pp. 15–31. North-Holland, Amsterdam (1975)

  28. Gehrke, M.: Generalized Kripke frames. Studia Logica 84, 241–275 (2006)

    MathSciNet  MATH  Google Scholar 

  29. Gehrke, M.: Canonical extensions, Esakia spaces, and universal models. In: Bezhanishvili, G. (ed.) Leo Esakia on Duality in Modal and Intuitionistic Logics, Outstanding Contributions to Logic, vol. 4, pp. 9–41. Springer, New York (2014)

    Google Scholar 

  30. Gehrke, M., van Gool, S.J.: Distributive envelopes and topological duality for lattices via canonical extensions. Order 31, 435–461 (2014)

    MathSciNet  MATH  Google Scholar 

  31. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 239, 345–371 (2001)

    MathSciNet  MATH  Google Scholar 

  32. Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Am. Math. Soc. 358, 573–590 (2006)

    MathSciNet  MATH  Google Scholar 

  33. Gehrke, M., Jansana, R., Palmigiano, A.: Canonical extensions for congruential logics with the deduction theorem. Ann. Pure Appl. Logic 161, 1502–1519 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Gehrke, M., Jansana, R., Palmigiano, A.: \(\Delta _1\)-completions of a poset. Order 30, 39–64 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Japonica 40(2), 207–215 (1994)

    MathSciNet  MATH  Google Scholar 

  36. Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Japonica 52(2), 197–213 (2000)

    MathSciNet  MATH  Google Scholar 

  37. Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94, 13–45 (2004)

    MathSciNet  MATH  Google Scholar 

  38. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Anna. Pure Appl. Logic 131(1–3), 65–102 (2005)

    MathSciNet  MATH  Google Scholar 

  39. Gehrke, M., Priestley, H.A.: Canonical extensions of double quasioperator algebras: An algebraic perspective on duality for certain algebras with binary operations. J. Pure Appl. Algebra 209(1), 269–290 (2007)

    MathSciNet  MATH  Google Scholar 

  40. Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Studia Logica 86(1), 31–68 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Gehrke, M., Priestley, H.A.: Canonical extensions and completions of posets and lattices. Rep. Math. Logic 43, 133–152 (2008)

    MathSciNet  MATH  Google Scholar 

  42. Gehrke, M., Vosmaer, J.: A view of canonical extension. In: Bezhanishvili, N. (ed.) Logic, Language, and Computation, TbiLLC 2009. Lecture Notes in Artificial Intelligence, vol. 6618, pp. 77–100. Springer, New York (2009)

    Google Scholar 

  43. Gehrke, M., Vosmaer, J.: Canonical extensions and canonicity via dcpo presentations. Theor. Comput. Sci. 412(25), 2714–2723 (2011)

    MathSciNet  MATH  Google Scholar 

  44. Goldblatt, R.: Metamathematics of modal logic. Ph.D. thesis, Victoria University, Wellington (1974). Included in [48]

  45. Goldblatt, R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)

    MathSciNet  MATH  Google Scholar 

  46. Goldblatt, R.: On closure under canonical embedding algebras. In: Andréka, H., Monk, J., Németi, I. (eds.) Algebraic Logic, Colloquia Mathematica Societatis János Bolyai, vol. 54, pp. 217–229. North-Holland, Amsterdam (1991)

    Google Scholar 

  47. Goldblatt, R.: Logics of Time and Computation, second edn. CSLI Lecture Notes No. 7. CSLI Publications, Stanford University (1992)

  48. Goldblatt, R.: Mathematics of Modality. CSLI Lecture Notes No. 43. CSLI Publications, Stanford University (1993)

  49. Goldblatt, R.: Elementary generation and canonicity for varieties of Boolean algebras with operators. Algebra Universalis 34, 551–607 (1995)

    MathSciNet  MATH  Google Scholar 

  50. Goldblatt, R.: Fine’s theorem on first-order complete modal logics. arXiv:1604.02196 (2016)

  51. Goldblatt, R., Hodkinson, I.: The McKinsey–Lemmon logic is barely canonical. Aust. J. Logic 5, 1–19 (2007). https://ojs.victoria.ac.nz/ajl/article/view/1783

  52. Goldblatt, R., Hodkinson, I., Venema, Y.: On canonical modal logics that are not elementarily determined. Logique et Analyse 181, 77–101 (2003). Published October 2004

    MathSciNet  MATH  Google Scholar 

  53. Goldblatt, R., Hodkinson, I., Venema, Y.: Erdős graphs resolve Fine’s canonicity problem. Bull. Symb. Logic 10(2), 186–208 (2004)

    MATH  Google Scholar 

  54. González, L.J., Jansana, R.: A topological duality for posets. Algebra Universalis 76(4), 455–478 (2016)

    MathSciNet  MATH  Google Scholar 

  55. van Gool, S.J.: Duality and canonical extensions for stably compact spaces. Ann. Pure Appl. Logic 159, 341–359 (2012)

    MathSciNet  MATH  Google Scholar 

  56. Gouveia, M.J., Priestley, H.A.: Profinite completions and canonical extension of semilttice reducts of distributive lattices. Houston J. Math. 39, 1117–1136 (2013)

    MathSciNet  MATH  Google Scholar 

  57. Gouveia, M.J., Priestley, H.A.: Canonical extensions and profinite completions of semilattices and lattices. Order 31, 189–216 (2014)

    MathSciNet  MATH  Google Scholar 

  58. Harding, J.: Canonical completions of lattices and ortholattices. Tatra Mt. Math. Publ. 15, 89–96 (1998)

    MathSciNet  MATH  Google Scholar 

  59. Harding, J.: The free orthomodular lattice on countably many generators is a subalgebra of the free orthomodular lattice on three generators. Algebra Universalis 48, 171–182 (2002)

    MathSciNet  MATH  Google Scholar 

  60. Harding, J.: On profinite completions and canonical extensions. Algebra Universalis 55, 293–296 (2006)

    MathSciNet  MATH  Google Scholar 

  61. Hartung, G.: A topological representation of lattices. Algebra Universalis 29, 273–299 (1992)

    MathSciNet  MATH  Google Scholar 

  62. Haviar, M., Priestley, H.A.: Canonical extensions of Stone and double Stone algebras: the natural way. Math. Slovaca 56, 53–78 (2006)

    MathSciNet  MATH  Google Scholar 

  63. Henkin, L.: The completeness of the first-order functional calculus. J. Symb. Logic 14, 159–166 (1949)

    MathSciNet  MATH  Google Scholar 

  64. Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras I. North-Holland, Amsterdam (1971)

    MATH  Google Scholar 

  65. Herrmann, C.: A finitely generated modular ortholattice. Can. Math. Bull. 24(2), 241–243 (1981)

    MathSciNet  MATH  Google Scholar 

  66. Hodkinson, I., Venema, Y.: Canonical varieties with no canonical axiomatisation. Trans. Am. Math. Soc. 357(11), 4579–4605 (2005)

    MathSciNet  MATH  Google Scholar 

  67. Hughes, G.E., Cresswell, M.J.: K1.1 is not canonical. Bulletin of the Section of Logic. Pol. Acad. Sci. 11, 109–112 (1982)

    MATH  Google Scholar 

  68. Jónsson, B.: Canonical extensions of bounded distributive lattice expansions. Abstract and notes for an invited talk at the International Conference on Order, Algebra and Logics, Vanderbilt University, June 2007. www.math.vanderbilt.edu/~oal2007/submissions/Jonsson.pdf

  69. Jónsson, B.: The role of universal algebra and lattice theory. Abstract of an invited talk at the 1992 New Zealand Mathematics Colloquium, Victoria University of Wellington

  70. Jónsson, B.: A survey of Boolean algebras with operators. Algebras and Orders. NATO ASI Series, vol. 389, pp. 239–286. Kluwer Academic Publishers, Norwell (1993)

    MATH  Google Scholar 

  71. Jónsson, B.: On the canonicity of Sahlqvist identities. Studia Logica 53, 473–491 (1994)

    MathSciNet  MATH  Google Scholar 

  72. Jónsson, B.: The preservation theorem for canonical extensions of Boolean algebras with operators. In: K.A. Baker, R. Wille (eds.) Lattice Theory and its Applications. Celebration of Garrett Birkhoff’s 80th Birthday. Research and Expositions in Mathematics, vol. 23, pp. 121–130. Heldermann Verlag, Berlin (1995)

    Google Scholar 

  73. Jónsson, B., Tarski, A.: Boolean algebras with operators. Bull. Am. Math. Soc. 54, 79–80 (1948)

    Google Scholar 

  74. Jónsson, B., Tarski, A.: Boolean algebras with operators, part I. Am. J. Math. 73, 891–939 (1951)

    MATH  Google Scholar 

  75. Jónsson, B., Tarski, A.: Boolean algebras with operators, part II. Am. J. Math. 74, 127–162 (1952)

    MATH  Google Scholar 

  76. Kalmbach, G.: Orthomodular Lattices. Academic Press, Cambridge (1983)

    MATH  Google Scholar 

  77. Kikot, S.: A dichotomy for some elementarily generated modal logics. Studia Logica 103(5), 1063–1093 (2015)

    MathSciNet  MATH  Google Scholar 

  78. Kotas, J.: An axiom system for the modular logic. Studia Logica 21, 17–38 (1967)

    MathSciNet  MATH  Google Scholar 

  79. Lemmon, E.J.: An Introduction to Modal Logic, American Philosophical Quarterly Monograph Series, vol. 11. Basil Blackwell, Oxford (1977). (Written in 1966 in collaboration with Dana Scott. Edited by Krister Segerberg)

  80. Lemmon, E.J., Scott, D.: Intensional logic (1966). Preliminary draft of initial chapters by E. J. Lemmon, Stanford University (later published as [79])

  81. MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937)

    MathSciNet  MATH  Google Scholar 

  82. Makinson, D.C.: On some completeness theorems in modal logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 12, 379–384 (1966)

    MathSciNet  MATH  Google Scholar 

  83. McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. 1. Wadsworth & Brooks/Cole, Belmont (1987)

    MATH  Google Scholar 

  84. McKinsey, J.C.C., Tarski, A.: The algebra of topology. Ann. Math. 45, 141–191 (1944)

    MathSciNet  MATH  Google Scholar 

  85. Morton, W.: Canonical extensions of posets. Algebra Universalis 72, 167–200 (2014)

    MathSciNet  MATH  Google Scholar 

  86. Moshier, M.A., Jipsen, P.: Topological duality and lattice expansions, I: A topological construction of canonical extensions. Algebra Universalis 71(2), 109–126 (2014)

    MathSciNet  MATH  Google Scholar 

  87. Moshier, M.A., Jipsen, P.: Topological duality and lattice expansions, II: Lattice expansions with quasioperators. Algebra Universalis 71(3), 221–234 (2014)

    MathSciNet  MATH  Google Scholar 

  88. Priestley, H.A.: Representations of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)

    MathSciNet  MATH  Google Scholar 

  89. de Rijke, M., Venema, Y.: Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras. Studia Logica 54, 61–78 (1995)

    MathSciNet  MATH  Google Scholar 

  90. Sahlqvist, H.: Completeness and correspondence in the first and second order semantics for modal logic. In: S. Kanger (ed.) Proceedings of the Third Scandinavian Logic Symposium, pp. 110–143. North-Holland, Amsterdam (1975)

  91. Segerberg, K.: Decidability of S4.1. Theoria 34, 7–20 (1968)

    MathSciNet  Google Scholar 

  92. Segerberg, K.: An Essay in Classical Modal Logic, Filosofiska Studier, vol. 13. Uppsala Universitet (1971)

  93. Smoryński, C.: Fixed point algebras. Bull. Am. Math. Soc. 6, 317–356 (1982)

    MathSciNet  MATH  Google Scholar 

  94. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)

    MathSciNet  MATH  Google Scholar 

  95. Suzuki, T.: Canonicity results of substructural and lattice-based logics. Rev. Symb. Logic 4, 1–42 (2011)

    MathSciNet  MATH  Google Scholar 

  96. Suzuki, T.: On canonicity of poset expansions. Algebra Universalis 66, 243–276 (2011)

    MathSciNet  MATH  Google Scholar 

  97. Thomason, S.K.: Semantic analysis of tense logic. J. Symb. Logic 37, 150–158 (1972)

    MathSciNet  MATH  Google Scholar 

  98. Urquhart, A.: A topological representation theory for lattices. Algebra Universalis 8, 45–58 (1978)

    MathSciNet  MATH  Google Scholar 

  99. Vosmaer, J.: Logic, algebra and topology: Investigations into canonical extensions, duality theory and point-free topology. Ph.D. thesis, Institute for Logic, Language and Computation, Universiteit van Amsterdam (2010). ILLC Dissertation Series DS-2010-10

Download references

Acknowledgements

The author thanks Mai Gehrke and Ian Hodkinson for some very helpful comments, information and improvements.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Victoria University of Wellington, Wellington, New Zealand

    Robert Goldblatt

Authors
  1. Robert Goldblatt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Goldblatt.

Additional information

In memoriam Bjarni Jónsson.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Goldblatt, R. Canonical extensions and ultraproducts of polarities. Algebra Univers. 79, 80 (2018). https://doi.org/10.1007/s00012-018-0562-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-018-0562-4

Mathematics Subject Classification

Keywords

Search

Navigation