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All Splitting Logics in the Lattice NExt(KTB)

  • Tomasz KowalskiEmail author
  • Yutaka Miyazaki
Part of the Trends in Logic book series (TREN, volume 28)

Abstract

It is proved that there are only two logics that split the lattice Next(KTB). The proof is based on the general splitting theorem by Kracht and conducted by a graph theoretic argument.

Keywords

splitting lattice of modal logics KTB 

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References

  1. [1]
    Blok, W., On the degree of incompleteness in modal logics and the covering relation in the lattice of modal logics, Tech. Rep. 78-07, Department of Mathematics, University of Amsterdam, 1978. Google Scholar
  2. [2]
    Blok, W., ‘The lattice of modal logics: an algebraic investigation’, Journal of Symbolic Logic, 45: 221–236, 1980. zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Blok, W., Köhler, P., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences II’, Algebra Universalis, 18: 334–379. Google Scholar
  4. [4]
    Blok, W., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences I’, Algebra Universalis, 15: 195–227. Google Scholar
  5. [5]
    Blok, W., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences III’, Algebra Universalis, 32: 545–608. Google Scholar
  6. [6]
    Blok, W., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences IV’, Algebra Universalis, 31: 1–35. Google Scholar
  7. [7]
    Burris, S.N., Sankappanavar, H.P., A Course in Universal Algebra, Springer Verlag, Berlin, 1981. zbMATHGoogle Scholar
  8. [8]
    Chagrov, A., Zakharyaschev, M., Modal Logic, Clarendon Press, Oxford. Google Scholar
  9. [9]
    Day, A., ‘Splitting lattices generate all lattices’, Algebra Universalis, 7: 163–169. Google Scholar
  10. [10]
    Fine, K., ‘An incomplete logic containing S4’, Theoria, 40: 23–29, 1974. zbMATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    Jankov, V.A., ‘The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures’, Soviet Mathematics Doklady, 4: 1203–1204. Google Scholar
  12. [12]
    Kowalski, T., An outline of a topography of tense logics, Ph.D. thesis, Jagiellonian University, Kraków, 1997. Google Scholar
  13. [13]
    Kowalski, T., Ono, H., ‘Splittings in the variety of residuated lattices’, Algebra Universalis, 44: 283–298. Google Scholar
  14. [14]
    Kracht, M., ‘Even more about the lattice of tense logics’, Archive of Mathematical Logic, 31: 243–357. Google Scholar
  15. [15]
    Kracht, M., Tools and Techniques in Modal Logic, Studies in Logics, vol. 42, Elsevier, Amsterdam. Google Scholar
  16. [16]
    Kracht, M., ‘An almost general splitting theorem for modal logic’, Studia Logica, 49: 455–470, 1990. zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    Makinson, D.C., ‘Some embedding theorems for modal logic’, Notre Dame Journal of Formal Logic, 12: 252–254, 1971. zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    McKenzie, R., ‘Equational bases and non-modular lattice varieties’, Transactions of the American Mathematical Society, 156: 1–43. Google Scholar
  19. [19]
    Miyazaki, Y., ‘A splitting logic in NEXT(KTB)’, Studia Logica, 85: 399–412, 2007. Google Scholar
  20. [20]
    Miyazaki, Y., ‘Kripke incomplete logics containing KTB’, Studia Logica, 85: 311–326, 2007. Google Scholar
  21. [21]
    Rautenberg, W., ‘Splitting lattices of logics’, Archiv für Mathematische Logik, 20: 155–159. Google Scholar
  22. [22]
    Stevens, M., Kowalski T., Minimal varieties of KTB-algebras, manuscript. Google Scholar
  23. [23]
    Whitman, Ph.M., ‘Splittings of a lattice’, American Journal of Mathematics, 65: 179–196. Google Scholar
  24. [24]
    Wolter, F., Lattices of modal logics, Ph.D. thesis, Freie Universität, Berlin, 1993. Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Research School of Information Sciences and EngineeringAustralian National UniversityCanberraAustralia
  2. 2.Meme Media LaboratoryHokkaido UniversitySapporoJAPAN

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