Search engine for discovering works of Art, research articles, and books related to Art and Culture
Javascript must be enabled to continue!
Representation and duality for Hilbert algebras
View through CrossRef
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.
Title: Representation and duality for Hilbert algebras
Description:
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.
Related Results
Finitely Presented Heyting Algebras
Finitely Presented Heyting Algebras
In this paper we study the structure of finitely presented Heyting<br />algebras. Using algebraic techniques (as opposed to techniques from proof-theory) we show that every s...
Differential graded vertex Lie algebras
Differential graded vertex Lie algebras
This is the continuation of the study of differential graded (dg) vertex algebras defined in our previous paper [Caradot et al., “Differential graded vertex operator algebras and t...
Relations between L-algebras and other logical algebras
Relations between L-algebras and other logical algebras
In this paper, by considering the notion of L-algebra, we show that there are relations between L-algebras and some of other logical algebras such as residuated lattices, MTL-alge...
Quantum B-algebras
Quantum B-algebras
Abstract
The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic...
On Kreb Algebras
On Kreb Algebras
In this paper, kreb algebras are introduced. It is shown that that the class of kreb algebras is a wider class than the class of BCI algebras. Properties of kreb algebras are prese...
Weak pseudo-BCK algebras
Weak pseudo-BCK algebras
Abstract
In this paper we define and study the weak pseudo-BCK algebras as generalizations of weak BCK-algebras, extending some results given by Cı⃖rulis for weak BC...
On FBZ-Algebras
On FBZ-Algebras
This paper introduces the concept of FBZ-algebra as a generalization of fuzzy implication algebra and investigates its fundamental properties. We establish a sufficient condition f...
Malcev Yang-Baxter equation, weighted $\mathcal{O}$-operators on Malcev algebras and post-Malcev algebras
Malcev Yang-Baxter equation, weighted $\mathcal{O}$-operators on Malcev algebras and post-Malcev algebras
The purpose of this paper is to study the $\mathcal{O}$-operators on Malcev algebras and discuss the solutions of Malcev Yang-Baxter equation by $\mathcal{O}$-operators. Furthe...