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Topological Representation of Intuitionistic and Distributive Abstract Logics

Abstract

We continue work of our earlier paper (Lewitzka and Brunner in Log Univers 3(2):219–241, 2009) where abstract logics and particularly intuitionistic abstract logics are studied. Abstract logics can be topologized in a direct and natural way. This facilitates a topological study of classes of concrete logics whenever they are given in abstract form. Moreover, such a direct topological approach avoids the often complex algebraic and lattice-theoretic machinery usually applied to represent logics. Motivated by that point of view, we define in this paper the category of intuitionistic abstract logics with stable logic maps as morphisms, and the category of implicative spectral spaces with spectral maps as morphisms. We show the equivalence of these categories and conclude that the larger categories of distributive abstract logics and distributive sober spaces are equivalent, too.

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Acknowledgements

We would like to thank the anonymous referee for many helpful comments and suggestions improving this work. The first author would like to thank the support from MaToMUVI Project with Number 247584 and the hospitality of University of Salerno (UNISA), where a part of this work was finished in the fall of 2014. The second author thanks the hospitality of the University of Potsdam, where part of this work was finished in the fall of 2016.

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Correspondence to Steffen Lewitzka.

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Brunner, A.B.M., Lewitzka, S. Topological Representation of Intuitionistic and Distributive Abstract Logics. Log. Univers. 11, 153–175 (2017). https://doi.org/10.1007/s11787-017-0166-3

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  • DOI: https://doi.org/10.1007/s11787-017-0166-3

Keywords

  • Abstract logics
  • topology
  • spectral spaces with implication
  • duality
  • intuitionistic logics

Mathematics Subject Classification

  • Primary 03B22
  • 03G10
  • Secondary 03B20