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.
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References
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.
Blok, W., ‘The lattice of modal logics: an algebraic investigation’, Journal of Symbolic Logic, 45: 221–236, 1980.
Blok, W., Köhler, P., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences II’, Algebra Universalis, 18: 334–379.
Blok, W., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences I’, Algebra Universalis, 15: 195–227.
Blok, W., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences III’, Algebra Universalis, 32: 545–608.
Blok, W., Pigozzi, D., ‘On the structure of varieties with equationally definable principal congruences IV’, Algebra Universalis, 31: 1–35.
Burris, S.N., Sankappanavar, H.P., A Course in Universal Algebra, Springer Verlag, Berlin, 1981.
Chagrov, A., Zakharyaschev, M., Modal Logic, Clarendon Press, Oxford.
Day, A., ‘Splitting lattices generate all lattices’, Algebra Universalis, 7: 163–169.
Fine, K., ‘An incomplete logic containing S4’, Theoria, 40: 23–29, 1974.
Jankov, V.A., ‘The relationship between deducibility in the intuitionistic propositional calculus and finite implicational structures’, Soviet Mathematics Doklady, 4: 1203–1204.
Kowalski, T., An outline of a topography of tense logics, Ph.D. thesis, Jagiellonian University, Kraków, 1997.
Kowalski, T., Ono, H., ‘Splittings in the variety of residuated lattices’, Algebra Universalis, 44: 283–298.
Kracht, M., ‘Even more about the lattice of tense logics’, Archive of Mathematical Logic, 31: 243–357.
Kracht, M., Tools and Techniques in Modal Logic, Studies in Logics, vol. 42, Elsevier, Amsterdam.
Kracht, M., ‘An almost general splitting theorem for modal logic’, Studia Logica, 49: 455–470, 1990.
Makinson, D.C., ‘Some embedding theorems for modal logic’, Notre Dame Journal of Formal Logic, 12: 252–254, 1971.
McKenzie, R., ‘Equational bases and non-modular lattice varieties’, Transactions of the American Mathematical Society, 156: 1–43.
Miyazaki, Y., ‘A splitting logic in NEXT(KTB)’, Studia Logica, 85: 399–412, 2007.
Miyazaki, Y., ‘Kripke incomplete logics containing KTB’, Studia Logica, 85: 311–326, 2007.
Rautenberg, W., ‘Splitting lattices of logics’, Archiv für Mathematische Logik, 20: 155–159.
Stevens, M., Kowalski T., Minimal varieties of KTB-algebras, manuscript.
Whitman, Ph.M., ‘Splittings of a lattice’, American Journal of Mathematics, 65: 179–196.
Wolter, F., Lattices of modal logics, Ph.D. thesis, Freie Universität, Berlin, 1993.
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Kowalski, T., Miyazaki, Y. (2009). All Splitting Logics in the Lattice NExt(KTB). In: Makinson, D., Malinowski, J., Wansing, H. (eds) Towards Mathematical Philosophy. Trends in Logic, vol 28. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-9084-4_4
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DOI: https://doi.org/10.1007/978-1-4020-9084-4_4
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