One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups

Dietzel, Carsten; Rump, Wolfgang; Zhang, Xia

doi:10.1016/j.jalgebra.2019.02.012

Cited by 10 publications

(2 citation statements)

References 36 publications

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“…Indeed, it was shown in [42] that the multiplicative group of a brace and subbraces which are its Sylow's subgroups (which have cardinality p n ) determine immediately the whole brace. There is also a connection with l-groups [19].…”

Section: Introductionmentioning

confidence: 99%

On the passage from finite braces to pre-Lie rings

Smoktunowicz¹

2022

Preprint

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Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality p n , for sufficiently large p. Moreover, there is an injective mapping from the set of left nilpotent pre-Lie rings into the set of left nilpotent braces of cardinality p n for n + 1 < p. As an application, by using well known results about the correspondence between braces and Hopf-Galois extensions we use pre-Lie algebras to describe Hopf-Galois extensions. We also translate some results from brace theory to pre-Lie algebras. For the passage from pre-Lie rings to braces we use the same method as described in [41].

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Section: Introductionmentioning

confidence: 99%

On the passage from finite braces to pre-Lie rings

Smoktunowicz¹

2022

Preprint

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“…However, there are already various notions of 'quantum set theory' in the literature; e.g. [12,16,31,32,36,37]. Later on, we were told that the objects that we are studying in this article have recently been studied under the name 'orthogonality spaces' in [28,38,39].…”

Section: Introductionmentioning

confidence: 99%

Ortho-sets and Gelfand spectra

Ding¹,

Ng²

2021

J. Phys. A: Math. Theor.

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Motivated by the relation on quantum states induced by zero transition probability, we introduce the notion of ortho-set, which is a set equipped with a relation ≠q satisfying: x ≠q y implies both x ≠ y and y ≠q x. For an ortho-set, a canonical complete ortholattice is constructed. Conversely, every complete ortholattice comes from an ortho-set in this way. Hence, the theory of ortho-sets captures almost everything about quantum logics. For a quantum system modeled by the self-adjoint part B sa of a C*-algebra B, we also introduce a ‘semi-classical object’ called the Gelfand spectrum. It is the ortho-set, P(B), of pure states of B equipped with an ‘ortho-topology’, which is a collection of subsets of P(B), defined via a hull-kernel construction with respects to closed left ideals of B. We establish a generalization of the Gelfand theorem by showing that a bijection between the Gelfand spectra of two quantum systems that preserves the respective ortho-topologies is induced by a Jordan isomorphism between the self-adjoint parts of the underlying C*-algebras (i.e. an isomorphism of the quantum systems), when the underlying C*-algebras satisfy a mild condition.

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Structure groups of L-algebras and Hurwitz action

Rump

2022

Geom Dedicata

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One-sided orthogonality, orthomodular spaces, quantum sets, and a class of Garside groups

Cited by 10 publications

References 36 publications

On the passage from finite braces to pre-Lie rings

On the passage from finite braces to pre-Lie rings

Ortho-sets and Gelfand spectra

Structure groups of L-algebras and Hurwitz action

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