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2009 | 7 | 3 | 463-478
Tytuł artykułu

Representation and duality for Hilbert algebras

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EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
7
Numer
3
Strony
463-478
Opis fizyczny
Daty
wydano
2009-09-01
online
2009-08-12
Twórcy
Bibliografia
  • [1] Balbes R., Dwinger Ph., Distributive lattices, University of Missouri Press, 1974
  • [2] Busneag D., A note on deductive systems of a Hilbert algebra, Kobe J. Math., 1985, 2, 29–35
  • [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
  • [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
  • [5] Celani S.A., Modal Tarski algebras, Reports on Mathematical Logic, 2005, 39, 113–126
  • [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
  • [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
  • [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
  • [9] Diego A., Sur les algèbres de Hilbert, Hermann, Paris, Collection de Logique Mathématique, Sér. A, 1966, 21 (in French)
  • [10] Koppelberg S., General theory of Boolean algebras, In: Monk D., Bonnet R. (Eds.), Handbook of Boolean Algebras, Vol. 1, North Holland, Amsterdam, 1989
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-009-0032-5
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