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Duality Theory: Hybrids

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Heyting Algebras

Part of the book series: Trends in Logic ((TREN,volume 50))

Abstract

This chapter covers the duality theory for closure algebras and Heyting algebras. The notion of a hybrid of topology and order is introduced, and the fundamental properties of hybrids are studied. It is proved that the category of hybrids and hybrid maps is dually equivalent to the category of closure algebras and closure algebra homomorphisms, while the category of strict hybrids and hybrid maps is dually equivalent to the category of Heyting algebras and Heyting homomorphisms. The notion of a Grzegorczyk algebra is introduced, and several characterizations of these algebras are given using the duality theory. From these characterizations it follows that the category of skeletal closure algebras is a subcategory of the category of Grzegorczyk algebras. Finally, it is proved that the variety of Grzegorczyk algebras is generated by its finite members, and some consequences of this result are derived.

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Notes

  1. 1.

    Editorial note: For the definition of the exponential (Vietoris) topology, see Definition 1.3.4.

  2. 2.

    Editorial note: Following the policy for omission of parentheses explained in the Preface, in contexts involving \({\mathbf {\mathsf{{C}}}}\), parentheses will sometimes be omitted after \(R\) and \(R^{-1}\) for readability.

  3. 3.

    Editorial note: The morphisms of the category \(\mathsf {HYB}\) will be introduced in Definition 3.3.1.

  4. 4.

    Editorial note: The existence of such a maximal chain follows from Proposition 1.4.10.

  5. 5.

    Editorial note: See, e.g., Example D on p. 26 of [18].

  6. 6.

    Editorial note: Since (X, R) is a chain, the family \(\{R(x) : x\in \mathrm {Gap}(X,R)\}\) is a ring of sets.

  7. 7.

    Editorial note: See, e.g., Corollary 3.1.14 of  R. Engelking, General Topology, 2nd ed. (Heldermann-Verlag), 1989.

  8. 8.

    Editorial note: This is a nonstandard use of the term ‘Dedekind complete’, which usually means that each nonempty subset bounded above has a supremum (resp. bounded below has an infimum).

  9. 9.

    Editorial note: A subset \(Z\) of \(X\) is upward (resp. downard) directed if for all \(x,y\in Z\), there is \(z\in Z\) such that \(xRz\) and \(yRz\) (resp. \(zRx\) and \(zRy\)).

  10. 10.

    Editorial note: See Proposition 5.9 on p. 33 of [3].

  11. 11.

    Editorial note: This result is now known as Esakia’s lemma ; see, e.g., p. 350 of A. Chagrov and M. Zakharyaschev, Modal Logic (Clarendon Press), 1997. Note that the lemma does not require that \(R\) be reflexive or transitive.

  12. 12.

    Editorial note: It is straightforward to see that \((X_i,\Omega _i,R_i)^*=(B(X_i),R_i^{-1})\) is indeed a closure algebra.

  13. 13.

    Editorial note: Indeed, it is easy to see that f being surjective implies that \(f^{-1}\) is injective. For the converse, given \(y\in Y\), we have that \(\{y\}\) is the intersection of the downward directed family of clopens \(A_i\) containing y. Since each \(A_i\) is nonempty, and \(f^{-1}\) is injective, \(f^{-1}(A_i)\) is nonempty. Thus, by compactness, the intersection of the \(f^{-1}(A_i)\) is nonempty, so \(f^{-1}(\{y\})\) is nonempty. This shows that f is surjective.

    To see that \(f^{-1}\) being surjective implies that f is injective, use the fact that distinct points are distinguished by clopens. For the converse, suppose A is a clopen subset of X. Then f(A) is closed in Y, so f(A) is the intersection of clopens \(B_i\) such that \(f(A)\subseteq B_i\). Thus, \(f^{-1}(f(A))=\bigcap \{f^{-1}(B_i)\mid i\in I\}\), and since f is injective, \(A=f^{-1}(f(A))\), so \(A=\bigcap \{f^{-1}(B_i)\mid i\in I\}\). Using compactness, it follows that there is one clopen B such that \(A=f^{-1}(B)\).

    Note that Lemmas 3.3.13(3) and 3.3.14(3) are inter-derivable using the Duality Theorem (the proof of which does not require these lemmas).

  14. 14.

    Editorial note: That \((\cdot )_*\) is a well-defined contravariant functor follows from Lemmas 3.3.11 and 3.3.13 (and the easily verifiable facts that it preserves identity and composition), and that \((\cdot )^*\) is a well-defined contravariant functor follows from Lemma 3.3.14 (and the same easily verifiable facts). That \((\cdot )_*\) satisfies part (a) of the definition of coequivalence (Definition (2)) follows from Lemma 3.3.15. That \((\cdot )_*\) satisfies part (b) is a consequence of Lemmas 3.3.13 and 3.3.14

  15. 15.

    Editorial note: It is easy to see that \((X_0,\Omega _0)\) is a zero-dimensional Hausdorff space.

  16. 16.

    Editorial note: Zorn’s lemma states that a partially ordered set in which every chain has an upper bound has a maximal element (see, e.g., [11]).

  17. 17.

    Editorial note: The proof is similar to that of the analogous part of Lemma 3.3.13(3).

  18. 18.

    Editorial note: Since \((X/E, \Omega _E)\) is compact and distinct points are separated by clopens, it follows that \((X/E, \Omega _E)\) is zero-dimensional. See, e.g., p. 69 of P. Johnstone, Stone Spaces (Cambridge University Press), 1982.

  19. 19.

    Editorial note: The Gödel translation  T is defined as follows: if p is a propositional variable, let \(T(p)=\Box p\), and for arbitrary formulae p, q, let \(T(p\vee q)=T(p)\vee T(q)\), \(T(p\wedge q)=T(p)\wedge T(q)\), \(T(p\rightarrow q)=\Box (T(p)\rightarrow T(q))\), and \(T(\lnot p)=\Box \lnot T(p)\).

  20. 20.

    Editorial note: The superintuitionistic fragment of \(\sigma \) is the superintuitionistic logic \({\{p\mid T(p)\in \sigma \}}\).

  21. 21.

    Editorial note: For a proof, see Corollary 9.64 (and Exercise 9.21) in A. Chagrov and M. Zakharyaschev, Modal Logic (Clarendon Press), 1997.

  22. 22.

    Editorial note: S5 is obtained by adding to S4 the axiom \(\Diamond p\rightarrow \Box \Diamond p\).

  23. 23.

    Editorial note: This result is now known as the Blok–Esakia Theorem; see p. 325 of A. Chagrov and M. Zakharyaschev, Modal Logic (Clarendon Press), 1997.

  24. 24.

    Editorial note: Lemma  3.5.11 is an algebraic formulation of Theorem 3.5.2 provided that the rule (R3) is reformulated in terms of \(\Diamond \).

  25. 25.

    Editorial note: Julia Ilin pointed out a gap in Esakia’s original proof, so we have replaced it by an algebraic version of the proof from pp. 158–9 of G. Boolos, The Logic of Provability (Cambridge University Press), 1993.

  26. 26.

    Editorial note: A logic is finitely approximable  if it is characterized (in the sense of Sect. 2.3.3) by a class of finite algebras.

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Authors and Affiliations

  1. Tbilisi, Georgia

    Leo Esakia (Deceased)

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  1. Leo Esakia
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Editors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM, USA

    Guram Bezhanishvili

  2. Department of Philosophy and Group in Logic and the Methodology of Science, University of California, Berkeley, CA, USA

    Wesley H. Holliday

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Esakia, L. (2019). Duality Theory: Hybrids. In: Bezhanishvili, G., Holliday, W. (eds) Heyting Algebras. Trends in Logic, vol 50. Springer, Cham. https://doi.org/10.1007/978-3-030-12096-2_3

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