Representation and duality for Hilbert algebras

Sergio Celani; Leonardo Cabrer; Daniela Montangie

Open Mathematics (2009)

  • Volume: 7, Issue: 3, page 463-478
  • ISSN: 2391-5455

Abstract

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In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.

How to cite

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Sergio Celani, Leonardo Cabrer, and Daniela Montangie. "Representation and duality for Hilbert algebras." Open Mathematics 7.3 (2009): 463-478. <http://eudml.org/doc/269367>.

@article{SergioCelani2009,
abstract = {In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.},
author = {Sergio Celani, Leonardo Cabrer, Daniela Montangie},
journal = {Open Mathematics},
keywords = {Hilbert algebras; Representation theorem; Topological duality; Deductive systems; Hilbert algebra; representation theorem; topological duality; deductive system; category; Hilbert space},
language = {eng},
number = {3},
pages = {463-478},
title = {Representation and duality for Hilbert algebras},
url = {http://eudml.org/doc/269367},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Sergio Celani
AU - Leonardo Cabrer
AU - Daniela Montangie
TI - Representation and duality for Hilbert algebras
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 463
EP - 478
AB - In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.
LA - eng
KW - Hilbert algebras; Representation theorem; Topological duality; Deductive systems; Hilbert algebra; representation theorem; topological duality; deductive system; category; Hilbert space
UR - http://eudml.org/doc/269367
ER -

References

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  1. [1] Balbes R., Dwinger Ph., Distributive lattices, University of Missouri Press, 1974 Zbl0321.06012
  2. [2] Busneag D., A note on deductive systems of a Hilbert algebra, Kobe J. Math., 1985, 2, 29–35 Zbl0584.06005
  3. [3] Celani S.A., A note on homomorphism of Hilbert algebras, Int. J. Math. Math. Sci., 2002, 29(1), 55–61 http://dx.doi.org/10.1155/S0161171202011134 Zbl0993.03089
  4. [4] Celani S.A., Representation of Hilbert algebras and implicative Semilattices, Cent. Eur. J. Math., 2003, 1(4), 561–572 http://dx.doi.org/10.2478/BF02475182 Zbl1034.03056
  5. [5] Celani S.A., Modal Tarski algebras, Reports on Mathematical Logic, 2005, 39, 113–126 Zbl1105.03069
  6. [6] Celani S.A., Cabrer L.M., Duality for finite Hilbert algebras, Discrete Math., 2005, 305, 74–99 http://dx.doi.org/10.1016/j.disc.2005.09.002 Zbl1084.03050
  7. [7] Celani S.A., Cabrer L.M., Topological duality for Tarski algebras, Algebra Universalis, 2008, 58, 73–94 http://dx.doi.org/10.1007/s00012-007-2041-1 Zbl1136.03046
  8. [8] Chajda I., Halaš P., Zedník J., Filters and annihilators in implication algebras, Acta Universitatis Palackianae Olomucensis, Facultas Rerum Naturalium, Mathematica, 1998, 37, 41–45 Zbl0967.03059
  9. [9] Diego A., Sur les algèbres de Hilbert, Hermann, Paris, Collection de Logique Mathématique, Sér. A, 1966, 21 (in French) Zbl0144.00105
  10. [10] Koppelberg S., General theory of Boolean algebras, In: Monk D., Bonnet R. (Eds.), Handbook of Boolean Algebras, Vol. 1, North Holland, Amsterdam, 1989 Zbl0676.06019

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