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Easkia Duality and Its Extensions

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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

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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 \(Y\).

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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.

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Authors and Affiliations

  1. Facultad de Ciencias Exactas, Universidad Nacional del Centro, Pinto, 399, 7000, Tandil, Argentina

    Sergio A. Celani

  2. Department of Logic, History and Philosophy of Science, University of Barcelona, Montalegre 6, 08001, Barcelona, Spain

    Ramon Jansana

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  1. Sergio A. Celani
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Correspondence to Ramon Jansana .

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  1. New Mexico State University, Las Cruces, New Mexico, USA

    Guram Bezhanishvili

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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

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