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Representation and duality for Hilbert algebras

100%
Celani S., Cabrer L., Montangie D.
Open Mathematics
|
2009
|
tom 7
|
nr 3
463-478
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.
2
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Some properties of epimorphisms of Hilbert algebras

75%
Buşneag D., Ghiţă M.
Open Mathematics
|
2010
|
tom 8
|
nr 1
41-52
EN
This paper represents a start in the study of epimorphisms in some categories of Hilbert algebras. Even if we give a complete characterization for such epimorphisms only for implication algebras, the following results will make possible the construction of some examples of epimorphisms which are not surjective functions. Also, we will show that the study of epimorphisms of Hilbert algebras is equivalent with the study of epimorphisms of Hertz algebras.
3
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Representation of Hilbert algebras and implicative semilattices

63%
Celani S.
Open Mathematics
|
2003
|
tom 1
|
nr 4
561-572
EN
In this paper we shall give a topological representation for Hilbert algebras that extend the topological representation given by A. Diego in [4]. For implicative semilattices this representation gives a full duality. We shall also consider the representation for Boolean ring.
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