Abstract
A fundamental result from Boolean modal logic states that a first-order definable class of Kripke frames defines a logic that is validated by all of its canonical frames. We generalise this to the level of non-distributive logics that have a relational semantics provided by structures based on polarities. Such structures have associated complete lattices of stable subsets, and these have been used to construct canonical extensions of lattice-based algebras. We study classes of structures that are closed under ultraproducts and whose stable set lattices have additional operators that are first-order definable in the underlying structure. We show that such classes generate varieties of algebras that are closed under canonical extensions. The proof makes use of a relationship between canonical extensions and MacNeille completions.
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References
Birkhoff, G., Lattice Theory, first edn., American Mathematical Society, New York, 1940.
Blackburn, P., M. De Rijke, and Y. Venema, Modal Logic, Cambridge University Press, 2001.
Chang, C. C., and H. J. Keisler, Model Theory, North-Holland, Amsterdam, 1973.
Chernilovskaya, A., M. Gehrke, and L. Van Rooijen, Generalized Kripke semantics for the Lambek -Grishin calculus, Logic Journal of the IGPL 20(6): 1110–1132, 2012.
Conradie, W., S. Frittella, A. Palmigiano, M. Piazzai, A. Tzimoulis, and N. M. Wijnberg, Categories: How I learned to stop worrying and love two sorts, in J. Väänänen et al., (eds.), WoLLIC 2016, vol. 9803 of Lecture Notes in Computer Science, Springer-Verlag, 2016, pp. 145–164.
Conradie, W., and A. Palmigiano, Algorithmic correspondence and canonicity for non-distributive logics, 2016. arXiv:1603.08515.
Coumans, D., M. Gehrke, and L. Van Rooijen, Relational semantics for full linear logic, Journal of Applied Logic 12(1): 50–66, 2014.
Davey, B. A., and H. A. Priestley, Introduction to Lattices and Order, Cambridge University Press, RR 1990.
Dunn, J. M., M. Gehrke, and A. Palmigiano, Canonical extensions and relational completeness of some substructural logics, The Journal of Symbolic Logic 70(3): 713–740, 2005.
Fine, K., Some connections between elementary and modal logic, in S. Kanger, (ed.), Proceedings of the Third Scandinavian Logic Symposium, North-Holland, 1975, pp. 15–31.
Galatos, N., P. Jipsen, T. Kowalski, and H. Ono, Residuated Lattices : An Algebraic Glimpse at Substructural Logics, vol. 151 of Studies in Logic and the Foundations of Mathematics, Elsevier, 2007.
Gehrke, M., Generalized Kripke frames, Studia Logica 84: 241–275, 2006.
Gehrke, M., and J. Harding, Bounded lattice expansions, Journal of Algebra 239: 345–371, 2001.
Gehrke, M., J. Harding, and Y. Venema, MacNeille completions and canonical extensions, Transactions of the American Mathematical Society 358: 573–590, 2006.
Goldblatt, R., Varieties of complex algebras, Annals of Pure and Applied Logic 44: 173–242, 1989.
Goldblatt, R., Canonical extensions and ultraproducts of polarities, Algebra Universalis 79, 2018.
Goldblatt, R., Morphisms and duality for polarities and lattices with operators, 2019.
Goldblatt, R., Fine’s theorem on first-order complete modal logics, in M. Dumitru, (ed.), Metaphysics, Meaning and Modality. Themes from Kit Fine, Oxford University Press, 2020. Also arXiv:1604.02196.
Jónsson, B., and A. Tarski, Boolean algebras with operators, part I, American Journal of Mathematics 73: 891–939, 1951.
Theunissen, M., and Y. Venema, MacNeille completions of lattice expansions, Algebra Universalis 57: 143–193, 2007.
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Presented by Constantine Tsinakis
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Goldblatt, R. Definable Operators on Stable Set Lattices. Stud Logica 108, 1263–1280 (2020). https://doi.org/10.1007/s11225-020-09896-0
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DOI: https://doi.org/10.1007/s11225-020-09896-0