# 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

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topSergio 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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- [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] Celani S.A., Modal Tarski algebras, Reports on Mathematical Logic, 2005, 39, 113–126 Zbl1105.03069
- [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] 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] 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] Diego A., Sur les algèbres de Hilbert, Hermann, Paris, Collection de Logique Mathématique, Sér. A, 1966, 21 (in French) Zbl0144.00105
- [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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