Representation and duality for Hilbert algebras

• Autorzy: Sergio Celani , Leonardo Cabrer , Daniela Montangie
• Języki publikacji: 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.

Słowa kluczowe

EN

Hilbert algebras; Representation theorem; Topological duality; Deductive systems

Kategorie tematyczne

• 06E15: Stone spaces (Boolean spaces) and related structures
• 06F35: BCK-algebras, BCI-algebras
• 03G25: Other algebras related to logic

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2009
• Tom: 7
• Numer: 3
• Strony: 463-478

Daty

wydano

2009-09-01

online

2009-08-12

Twórcy

autor

Sergio Celani

• CONICET and Universidad Nacional del Centro

autor

Leonardo Cabrer

• CONICET and Universidad Nacional del Centro

autor

Daniela Montangie

• Universidad Nacional del Comahue

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

Identyfikatory

DOI

10.2478/s11533-009-0032-5