Skip to main content

Easkia Duality and Its Extensions

  • Chapter
  • First Online:
Leo Esakia on Duality in Modal and Intuitionistic Logics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 4))

Abstract

In recent years Esakia duality for Heyting algebras has been extended in two directions. First to weak Heyting algebras, namely distributive lattices with an implication with weaker properties than that of the implication of a Heyting algebra, and secondly to implicative semilattices. The first algebras correspond to subintuitionistic logics, the second ones to the conjunction and implication fragment of intuitionistic logic. Esakia duality has also been complemented with dualities for categories whose objects are Heyting algebras and whose morphisms are maps that preserve less structure than homomorphisms of Heyting algebras. In this chapter we survey these developments.

In memory of Leo Esakia

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The term ‘Priestley quasi-order’ is introduced in [6, 10].

  2. 2.

    Note that if \(L\) and \(L^\prime \) are bounded relatively pseudo-complemented distributive lattices (the residuation operation, or implication, is not part of the signature) and \(h: L \rightarrow L^\prime \) is a bounded lattice homomorphism, then we only have that for \(a, b \in L\), \(h(a \rightarrow b) \le h(a) \rightarrow ' h(b)\).

  3. 3.

    In [25] the maps between Esakia spaces that are order preserving and satisfy that for every \(x \in X_1\), \({\uparrow }f(x) \subseteq f[{\uparrow }x]\) are called strongly isotone. That is, they are the maps such that for every \(x \in X_1\), \({\uparrow }f(x) = f[{\uparrow }x]\).

  4. 4.

    The full subcategory with Esakia spaces as objects has as morphisms the continuous and order preserving maps between them, but not all of them satisfy that for every \(x \in X_1\), \({\uparrow }f(x) \subseteq f[{\uparrow }x]\).

  5. 5.

    In [16] they are called weakly Heyting, but weak Heyting appears to be a better terminology. It comes from [12].

  6. 6.

    Two algebras of different similarity type are term-wise definitionally equivalent if every principal operation of one is definable by a term of the other.

  7. 7.

    Halmos introduced the term ‘hemimorphism’ in the above sense, but in the literature we find ‘hemimorphism’ applied to the meet and top preserving maps as well, see e.g. [51].

  8. 8.

    If \(X = \langle X, \tau , \le \rangle \) is a Priestley space and \(Y \subseteq X\), \(\mathrm {max}(Y)\) denotes the set of \(\le \)-maximal elements of \(Y\).

References

  1. Adams M (1973) The Frattini sublattice of a distributive lattice. Algebra Universalis 3:216–228

    Google Scholar 

  2. Adams M (1986) Maximal subalgebras of Heyting algebras. Proc Edinburgh Math Soc 29:359–365

    Google Scholar 

  3. Ardeshir M, Ruitenburg W (1998) Basic propositional calculus I. Math Logic Q 44:317–343

    Article  Google Scholar 

  4. Balbes R, Dwinger P (1974) Distributive lattices. University of Missouri Press, Columbia

    Google Scholar 

  5. Bezhanishvili G, Bezhanishvili N (2009) An algebraic approach to canonical formulas: intuitionistic case. Rev Symbolic Logic 2:517–549

    Article  Google Scholar 

  6. Bezhanishvili G, Bezhanishvili N, Gabelaia D, Kurz A (2010) Bitopological duality for distributive lattices and Heyting algebras. Math Struct Comput Sci 20:359–393

    Article  Google Scholar 

  7. Bezhanishvili G, Ghilardi S, Jibladze M (2011) An algebraic approach to subframe logics. Modal case. Notre Dame J Formal Logic 52:187–202

    Article  Google Scholar 

  8. Bezhanishvili G, Jansana R (2008) Duality for distributive and implicative semilattices. Preprints of University of Barcelona Research Group in Non-Classical Logics

    Google Scholar 

  9. Bezhanishvili G, Jansana R (2011) Priestley style duality for distributive meet-semilattices. Studia Logica 98:83–123

    Article  Google Scholar 

  10. Bezhanishvili G, Jansana R (2011) Generalized Priestley quasi-orders. Order 28:201–220

    Article  Google Scholar 

  11. Bezhanishvili G, Jansana R (2013) Esakia style duality for implicative semilattices. Appl Categorical Struct 21:181–208

    Google Scholar 

  12. Bezhanishvili N, Gehrke M (2011) Finitely generated free Heyting algebras via Birkhoff duality and coalgebra. Logical Methods Comput Sci 2(9):1–24

    Google Scholar 

  13. Celani S (2003) Representation of Hilbert algebras and implicative semilattices. Cent Eur J Math 1:561–572

    Article  Google Scholar 

  14. Celani S (2006) n-linear weakly Heyting algebras. Math Logic Q 52:404–416

    Google Scholar 

  15. Celani S, Jansana R (2003) A closer look at some subintuitionistic logics. Notre Dame J Formal Logic 42:225–255

    Google Scholar 

  16. Celani S, Jansana R (2005) Bounded distributive lattices with a strict implication. Math Logic Q 51:219–246

    Article  Google Scholar 

  17. Cignoli R (1991) Distributive lattice congruences and Priestley spaces. In: Actas del primer congreso Antonio Monteiro, Universidad Nacional del Sur, Bahía Blanca, pp 81–84

    Google Scholar 

  18. Cignoli R, Lafalce S, Petrovich A (1991) Remarks on Priestley duality for distributive lattices. Order 8:299–315

    Article  Google Scholar 

  19. Cornish W (1975) On H. Priestley’s dual of the category of bounded distributive lattices. Matematički Vesnik 12:329–332

    Google Scholar 

  20. Davey B, Galati J (2003) A coalgebraic view of Heyting duality. Studia Logica 75:259–270

    Google Scholar 

  21. Davey B, Priestley H (2002) Introduction to lattices and order, 2nd edn. Cambridge University Press, Cambridge

    Google Scholar 

  22. D\(\breve{\text{ o }}\)sen K (1993) Modal translations in \(K\) and \(D\). In: de Rijke M (ed) Diamonds and defaults. Kluwer Academic Publishers, Dordrecht, pp 103–127

    Google Scholar 

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

    Google Scholar 

  24. Epstein G, Horn A (1976) Logics which are characterized by subresiduated lattices. Zeitschrift fur Mathematiche Logik und Grundlagen der Mathematik 22:199–210

    Article  Google Scholar 

  25. Esakia L (1974) Topological Kripke models. Soviet Math Dokl 15:147–151

    Google Scholar 

  26. Frink O (1954) Ideals in partially ordered sets. Am Math Mon 61:223–234

    Article  Google Scholar 

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

    Article  Google Scholar 

  28. Gehrke M (2014) Canonical extensions, Esakia spaces, and universal models (this volume)

    Google Scholar 

  29. Gierz G, Hofmann K, Keimel K, Lawson J, Mislove M, Scott D (2003) Continuous lattices and domains, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge

    Google Scholar 

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

    Article  Google Scholar 

  31. Grätzer G (1998) General lattice theory, 2nd edn. Birkhauser, Bassel (New appendices by the author with Davey B, Freese R, Ganter B, Greferath M, Jipsen P, Priestley H, Rose H, Schmidt E, Schmidt S, Wehrung F, Wille R)

    Google Scholar 

  32. Halmos P (1955) Algebraic logic, I. Monadic Boolean algebras. Compositio Mathematica 12:217–249

    Google Scholar 

  33. Halmos P (1962) Algebraic logic. Chelsea Pub. Co., New York

    Google Scholar 

  34. Hansoul G (1996) Priestley duality for some subalgebra lattices. Studia Logica 56:133–149

    Article  Google Scholar 

  35. Hansoul G (2003) Priestley duality for distributive semilattices. Institut de Mathématique, Université de Liège, Preprint 97.011

    Google Scholar 

  36. Hansoul G, Poussart C (2008) Priestley duality for distributive semilattices. Bull Soc Roy Sci Liège 77:104–119

    Google Scholar 

  37. Hansoul G, Vrancken-Mawet L (1987) The subalgebra lattice of a Heyting algebra. Czech Math J 37:34–41

    Google Scholar 

  38. Hochster M (1969) Prime ideal structure in commutative rings. Trans Am Math Soc 142:43–60

    Article  Google Scholar 

  39. Johnstone P (1982) Stone spaces. Cambridge University Press, Cambridge

    Google Scholar 

  40. Köhler P (1981) Brouwerian semilattices. Trans Am Math Soc 268:126

    Article  Google Scholar 

  41. Köhler P, Pigozzi D (1980) Varieties with equationally definable principal congruences. Algebra Universalis 11:213–219

    Article  Google Scholar 

  42. Kupke C, Kurz A, Venema Y (2004) Stone coalgebras. Theor Comput Sci 327:109–134

    Article  Google Scholar 

  43. Minari P (1986) Intermediate logics with the same disjunctionless fragment as intuitionistic logic. Studia Logica 45:207–222

    Article  Google Scholar 

  44. Monteiro L (1963) Axiomes indépendants pour les algèbres de Łukasiewicz trivalentes. Bulletin de la Societé des Sciences Mathématiques et Physiques de la R. P. Roumanie, Nouvelle Série 7, pp 199–202

    Google Scholar 

  45. Nemitz W (1965) Implicative semilattices. Trans Am Math Soc 117:128–142

    Google Scholar 

  46. Petrovich A (1996) Distributive lattices with an operator. Studia Logica 56:205–224

    Article  Google Scholar 

  47. Priestley H (1970) Representation of distributive lattices by means of ordered Stone spaces. Bull Lond Math Soc 2:186–190

    Google Scholar 

  48. Priestley H (1972) Ordered topological spaces and the representation of distributive lattices. Proc Lond Math Soc 24(3):507–530

    Google Scholar 

  49. Priestley H (1974) Stone lattices: a topological approach. Fundamenta Mathematicae 84:127–143

    Google Scholar 

  50. Rautenberg W (1979) Klassiche und nichtklassiche Aussagenlogik. Vieweg & Sohn, Braunschweig, Fried

    Google Scholar 

  51. Sambin G, Vaccaro V (1988) Topology and duality in modal logic. Ann Pure Appl Logic 37:249–296

    Article  Google Scholar 

  52. Stone M (1937) Topological representations of distributive lattices and Brouwerian logics. \(\check{\text{ C }}\)asopis pro pesto\(\acute{\text{ v }}\)van\(\acute{\text{ y }}\) matematiky a fysiky 67:1–25

    Google Scholar 

  53. Schmid J (2002) Quasiorders and sublattices of distributive lattices. Order 19:11–13

    Article  Google Scholar 

  54. Venema Y (2007) Algebras and coalgebras. In: Blackburn P, van Benthem J, Wolter F (eds) Handbook of Modal Logic, vol 3 of Studies in Logic and Practical Reasoning. Elsevier, Amsterdam, pp 331–426

    Google Scholar 

  55. Visser A (1981) Aspects of diagonalization and provability. PhD thesis, University of Utrecht, Utrecht

    Google Scholar 

  56. Visser A (1981) A propositional logic with explicit fixed points. Studia Logica 40:155–175

    Article  Google Scholar 

  57. Vrancken-Mawet L (1982) Le lattis des sous-algèbres d’un algèbre de Heyting finie. Bull Soc Rot Sci Liège 51:82–94

    Google Scholar 

  58. Vrancken-Mawet L (1984) The lattice of R-subalgebras of a bounded distributive lattice. Comment Math Univ Carolinae 25:1–17

    Google Scholar 

  59. Wright F (1957) Some remarks on Boolean duality. Portugaliae Mathematicae 16:109–117

    Google Scholar 

  60. Zakharyaschev M (1983) On intermediate logics. Soviet Mathematics Doklady 27(2):274–277

    Google Scholar 

  61. Zakharyaschev M (1987) The disjunction property of superintuitionistic and modal logics. Matematicheskie Zametki 42:729–738, 763

    Google Scholar 

  62. Zakharyaschev M (1989) Syntax and semantics of superintuitionistic logics. Algebra Logic 28:262–282

    Google Scholar 

Download references

Acknowledgments

We would like to thank an anonymous referee for the useful comments on the presentation of the chapter and also Guram Bezhanishvili for his useful comments and his careful reading of the chapter that lead to the improvement of English and style. The work on the chapter has been possible thanks to the Marie Curie Actions-International Research Staff Exchange Scheme (IRSES) MaToMUVI-FP7-PEOPLE-2009-IRSES from the European Union. The second author has also been partially supported by the grants 2009SGR-1433 of the AGAUR of the Generalitat de Catalunya and MTM2011-25747 of the Spanish Ministerio de Ciencia e Innovación, which includes eu feder funds.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ramon Jansana .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Celani, S.A., Jansana, R. (2014). Easkia Duality and Its Extensions. In: Bezhanishvili, G. (eds) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8860-1_4

Download citation

Publish with us

Policies and ethics