Advertisement

Algebra universalis

, 79:80 | Cite as

Canonical extensions and ultraproducts of polarities

  • Robert Goldblatt
Article
  • 19 Downloads
Part of the following topical collections:
  1. In memory of Bjarni Jónsson

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.

Keywords

Canonical extension Canonical variety Lattice Completion Lattice-based algebra MacNeille completion Ultraproduct Polarity Galois connection Hanf number 

Mathematics Subject Classification

03G10 06B23 03C20 06A15 06D50 

Notes

Acknowledgements

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

References

  1. 1.
    Almeida, A.: Canonical extensions and relational representations of lattices with negation. Stud. Log. 91, 171–199 (2009)MathSciNetzbMATHGoogle Scholar
  2. 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. 3.
    Bell, J.L., Slomson, A.B.: Models and Ultraproducts. North-Holland, Amsterdam (1969)zbMATHGoogle Scholar
  4. 4.
    Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions and canonical extensions of Heyting algebras. Order 23, 143–161 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bezhanishvili, G., Mines, R., Morandi, P.J.: Topo-canonical completions of closure algebras and Heyting algebras. Algebra Universalis 58, 1–34 (2008)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Birkhoff, G.: Lattice Theory, 1st edn. American Mathematical Society, New York (1940)Google Scholar
  7. 7.
    Bruns, G., Roddy, M.: A finitely generated modular ortholattice. Can. Math. Bull. 35(1), 29–33 (1992)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bulian, J., Hodkinson, I.: Bare canonicity of representable cylindric and polyadic algebras. Anna. Pure Appl. Logic 164(9), 884–906 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra. Springer-Verlag, New York (1981)zbMATHGoogle Scholar
  10. 10.
    Burris, S., Werner, H.: Sheaf constructions and their elementary properties. Trans. Am. Math. Soc. 248, 269–309 (1979)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973)zbMATHGoogle Scholar
  12. 12.
    Chernilovskaya, A., Gehrke, M., van Rooijen, L.: Generalized Kripke semantics for the Lambek-Grishin calculus. Logic J. IGPL 20(6), 1110–1132 (2012)MathSciNetzbMATHGoogle Scholar
  13. 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. 14.
    Conradie, W., Palmigiano, A.: Algorithmic correspondence and canonicity for non-distributive logics. arXiv:1603.08515 (2016)
  15. 15.
    Coumans, D.: Generalising canonical extension to the categorical setting. Ann. Pure Appl. Logic 163(12), 1940–1961 (2012)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Coumans, D., Gehrke, M., van Rooijen, L.: Relational semantics for full linear logic. J. Appl. Logic 12(1), 50–66 (2014)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Craig, A.: Canonical extensions of bounded lattices and natural duality for default bilattices. Ph.D. thesis, University of Oxford (2012)Google Scholar
  18. 18.
    Craig, A., Haviar, M.: Reconciliation of approaches to the construction of canonical extensions of bounded lattices. Math. Slovaca 64(6), 1335–1356 (2014)MathSciNetzbMATHGoogle Scholar
  19. 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)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Cresswell, M.J.: A Henkin completeness theorem for T. Notre Dame J. Formal Logic 8, 186–190 (1967)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Davey, B.A., Haviar, M., Priestley, H.A.: Boolean topological distributive lattices and canonical extensions. Appl. Categorical Struct. 15, 225–241 (2007)MathSciNetzbMATHGoogle Scholar
  22. 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)MathSciNetzbMATHGoogle Scholar
  23. 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)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  25. 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)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Fine, K.: Logics containing K4. Part I. J. Symbol. Logic 39(1), 31–42 (1974)MathSciNetzbMATHGoogle Scholar
  27. 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)Google Scholar
  28. 28.
    Gehrke, M.: Generalized Kripke frames. Studia Logica 84, 241–275 (2006)MathSciNetzbMATHGoogle Scholar
  29. 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. 30.
    Gehrke, M., van Gool, S.J.: Distributive envelopes and topological duality for lattices via canonical extensions. Order 31, 435–461 (2014)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 239, 345–371 (2001)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Am. Math. Soc. 358, 573–590 (2006)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Gehrke, M., Jansana, R., Palmigiano, A.: Canonical extensions for congruential logics with the deduction theorem. Ann. Pure Appl. Logic 161, 1502–1519 (2010)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Gehrke, M., Jansana, R., Palmigiano, A.: \(\Delta _1\)-completions of a poset. Order 30, 39–64 (2013)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Japonica 40(2), 207–215 (1994)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gehrke, M., Jónsson, B.: Monotone bounded distributive lattice expansions. Math. Japonica 52(2), 197–213 (2000)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94, 13–45 (2004)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Anna. Pure Appl. Logic 131(1–3), 65–102 (2005)MathSciNetzbMATHGoogle Scholar
  39. 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)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Gehrke, M., Priestley, H.A.: Duality for double quasioperator algebras via their canonical extensions. Studia Logica 86(1), 31–68 (2007)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Gehrke, M., Priestley, H.A.: Canonical extensions and completions of posets and lattices. Rep. Math. Logic 43, 133–152 (2008)MathSciNetzbMATHGoogle Scholar
  42. 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. 43.
    Gehrke, M., Vosmaer, J.: Canonical extensions and canonicity via dcpo presentations. Theor. Comput. Sci. 412(25), 2714–2723 (2011)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Goldblatt, R.: Metamathematics of modal logic. Ph.D. thesis, Victoria University, Wellington (1974). Included in [48]Google Scholar
  45. 45.
    Goldblatt, R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)MathSciNetzbMATHGoogle Scholar
  46. 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. 47.
    Goldblatt, R.: Logics of Time and Computation, second edn. CSLI Lecture Notes No. 7. CSLI Publications, Stanford University (1992)Google Scholar
  48. 48.
    Goldblatt, R.: Mathematics of Modality. CSLI Lecture Notes No. 43. CSLI Publications, Stanford University (1993)Google Scholar
  49. 49.
    Goldblatt, R.: Elementary generation and canonicity for varieties of Boolean algebras with operators. Algebra Universalis 34, 551–607 (1995)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Goldblatt, R.: Fine’s theorem on first-order complete modal logics. arXiv:1604.02196 (2016)
  51. 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. 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 2004MathSciNetzbMATHGoogle Scholar
  53. 53.
    Goldblatt, R., Hodkinson, I., Venema, Y.: Erdős graphs resolve Fine’s canonicity problem. Bull. Symb. Logic 10(2), 186–208 (2004)zbMATHGoogle Scholar
  54. 54.
    González, L.J., Jansana, R.: A topological duality for posets. Algebra Universalis 76(4), 455–478 (2016)MathSciNetzbMATHGoogle Scholar
  55. 55.
    van Gool, S.J.: Duality and canonical extensions for stably compact spaces. Ann. Pure Appl. Logic 159, 341–359 (2012)MathSciNetzbMATHGoogle Scholar
  56. 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)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Gouveia, M.J., Priestley, H.A.: Canonical extensions and profinite completions of semilattices and lattices. Order 31, 189–216 (2014)MathSciNetzbMATHGoogle Scholar
  58. 58.
    Harding, J.: Canonical completions of lattices and ortholattices. Tatra Mt. Math. Publ. 15, 89–96 (1998)MathSciNetzbMATHGoogle Scholar
  59. 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)MathSciNetzbMATHGoogle Scholar
  60. 60.
    Harding, J.: On profinite completions and canonical extensions. Algebra Universalis 55, 293–296 (2006)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Hartung, G.: A topological representation of lattices. Algebra Universalis 29, 273–299 (1992)MathSciNetzbMATHGoogle Scholar
  62. 62.
    Haviar, M., Priestley, H.A.: Canonical extensions of Stone and double Stone algebras: the natural way. Math. Slovaca 56, 53–78 (2006)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Henkin, L.: The completeness of the first-order functional calculus. J. Symb. Logic 14, 159–166 (1949)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras I. North-Holland, Amsterdam (1971)zbMATHGoogle Scholar
  65. 65.
    Herrmann, C.: A finitely generated modular ortholattice. Can. Math. Bull. 24(2), 241–243 (1981)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Hodkinson, I., Venema, Y.: Canonical varieties with no canonical axiomatisation. Trans. Am. Math. Soc. 357(11), 4579–4605 (2005)MathSciNetzbMATHGoogle Scholar
  67. 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)zbMATHGoogle Scholar
  68. 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. 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 WellingtonGoogle Scholar
  70. 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)zbMATHGoogle Scholar
  71. 71.
    Jónsson, B.: On the canonicity of Sahlqvist identities. Studia Logica 53, 473–491 (1994)MathSciNetzbMATHGoogle Scholar
  72. 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. 73.
    Jónsson, B., Tarski, A.: Boolean algebras with operators. Bull. Am. Math. Soc. 54, 79–80 (1948)Google Scholar
  74. 74.
    Jónsson, B., Tarski, A.: Boolean algebras with operators, part I. Am. J. Math. 73, 891–939 (1951)zbMATHGoogle Scholar
  75. 75.
    Jónsson, B., Tarski, A.: Boolean algebras with operators, part II. Am. J. Math. 74, 127–162 (1952)zbMATHGoogle Scholar
  76. 76.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, Cambridge (1983)zbMATHGoogle Scholar
  77. 77.
    Kikot, S.: A dichotomy for some elementarily generated modal logics. Studia Logica 103(5), 1063–1093 (2015)MathSciNetzbMATHGoogle Scholar
  78. 78.
    Kotas, J.: An axiom system for the modular logic. Studia Logica 21, 17–38 (1967)MathSciNetzbMATHGoogle Scholar
  79. 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) Google Scholar
  80. 80.
    Lemmon, E.J., Scott, D.: Intensional logic (1966). Preliminary draft of initial chapters by E. J. Lemmon, Stanford University (later published as [79])Google Scholar
  81. 81.
    MacNeille, H.M.: Partially ordered sets. Trans. Am. Math. Soc. 42, 416–460 (1937)MathSciNetzbMATHGoogle Scholar
  82. 82.
    Makinson, D.C.: On some completeness theorems in modal logic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 12, 379–384 (1966)MathSciNetzbMATHGoogle Scholar
  83. 83.
    McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. 1. Wadsworth & Brooks/Cole, Belmont (1987)zbMATHGoogle Scholar
  84. 84.
    McKinsey, J.C.C., Tarski, A.: The algebra of topology. Ann. Math. 45, 141–191 (1944)MathSciNetzbMATHGoogle Scholar
  85. 85.
    Morton, W.: Canonical extensions of posets. Algebra Universalis 72, 167–200 (2014)MathSciNetzbMATHGoogle Scholar
  86. 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)MathSciNetzbMATHGoogle Scholar
  87. 87.
    Moshier, M.A., Jipsen, P.: Topological duality and lattice expansions, II: Lattice expansions with quasioperators. Algebra Universalis 71(3), 221–234 (2014)MathSciNetzbMATHGoogle Scholar
  88. 88.
    Priestley, H.A.: Representations of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)MathSciNetzbMATHGoogle Scholar
  89. 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)MathSciNetzbMATHGoogle Scholar
  90. 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)Google Scholar
  91. 91.
    Segerberg, K.: Decidability of S4.1. Theoria 34, 7–20 (1968)MathSciNetGoogle Scholar
  92. 92.
    Segerberg, K.: An Essay in Classical Modal Logic, Filosofiska Studier, vol. 13. Uppsala Universitet (1971)Google Scholar
  93. 93.
    Smoryński, C.: Fixed point algebras. Bull. Am. Math. Soc. 6, 317–356 (1982)MathSciNetzbMATHGoogle Scholar
  94. 94.
    Stone, M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)MathSciNetzbMATHGoogle Scholar
  95. 95.
    Suzuki, T.: Canonicity results of substructural and lattice-based logics. Rev. Symb. Logic 4, 1–42 (2011)MathSciNetzbMATHGoogle Scholar
  96. 96.
    Suzuki, T.: On canonicity of poset expansions. Algebra Universalis 66, 243–276 (2011)MathSciNetzbMATHGoogle Scholar
  97. 97.
    Thomason, S.K.: Semantic analysis of tense logic. J. Symb. Logic 37, 150–158 (1972)MathSciNetzbMATHGoogle Scholar
  98. 98.
    Urquhart, A.: A topological representation theory for lattices. Algebra Universalis 8, 45–58 (1978)MathSciNetzbMATHGoogle Scholar
  99. 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-2010-10Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsVictoria University of WellingtonWellingtonNew Zealand

Personalised recommendations