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.
@article{bwmeta1.element.doi-10_2478_s11533-009-0032-5,
author = {Sergio Celani and Leonardo Cabrer and Daniela Montangie},
title = {Representation and duality for Hilbert algebras},
journal = {Open Mathematics},
volume = {7},
year = {2009},
pages = {463-478},
zbl = {1184.03064},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0032-5}
}
Sergio Celani; Leonardo Cabrer; Daniela Montangie. Representation and duality for Hilbert algebras. Open Mathematics, Tome 7 (2009) pp. 463-478. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-009-0032-5/
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