Concrete Quantum Logics and Δ -Logics, States and Δ -States

Hroch, Michal; Pták, Pavel

doi:10.1007/s10773-017-3359-x

Cited by 3 publications

(1 citation statement)

References 17 publications

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“…Recently there has been an effort to soundly introduce and study the notion of a symmetric difference in orthomodular lattices and posets (see [ 3 , 4 , 8 , 9 , 12 ], etc.). In the relation to quantum axiomatics, the idea has been to enrich the “quantum logics” with a kind of a operation.…”

Section: Introduction and Basic Notionsmentioning

confidence: 99%

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Burešová

2023

Int J Theor Phys

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Considering the inference rules in generalized logics, J.C. Abbott arrives to the notion of orthoimplication algebra (see Abbott (1970) and Abbott (Stud. Logica. 2:173–177, XXXV)). We show that when one enriches the Abbott orthoimplication algebra with a falsity symbol and a natural $$\mathbb {XOR}$$ XOR -type operation, one obtains an orthomodular difference lattice as an enriched quantum logic (see Matoušek (Algebra Univers. 60:185–215, 2009)). Moreover, we find that these two structures endowed with the natural morphisms are categorically equivalent. We also show how one can introduce the notion of a state in the Abbott $$\mathbb {XOR}$$ XOR algebras strenghtening thus the relevance of these algebras to quantum theories.

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Section: Introduction and Basic Notionsmentioning

confidence: 99%

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Burešová

2023

Int J Theor Phys

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Quantum Logics Defined by Divisibility Conditions

2018

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On the Set-Representable Orthomodular Posets that are Point-Distinguishing

Burešová,

Pták

2023

Int J Theor Phys

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Let us denote by SOMP the class of all set-representable orthomodular posets and by PDSOMP those elements of SOMP in which any pair of points in the underlying set P can be distinguished by a set (i.e., (P, L) ∈ PDSOMP precisely when for any pair x, y ∈ P there is a set A ∈ L with x ∈ A and y / ∈ A). In this note we first construct, for each (P, L) ∈ SOMP, a point-distinguishing orthomodular poset that is isomorphic to (P, L). We show that by using a generalized form of the Stone representation technique we also obtain point-distinguishing representations of (P, L). We then prove that this technique gives us point-distinguishing representations on which all two-valued states are determined by points (all two-valued states are Dirac states). Since orthomodular posets may be regarded as abstract counterparts of event structures about quantum experiments, results of this work may have some relevance for the foundation of quantum mechanics.

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Concrete Quantum Logics and Δ -Logics, States and Δ -States

Cited by 3 publications

References 17 publications

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Generalized $$\mathbb {XOR}$$ Operation and the Categorical Equivalence of the Abbott Algebras and Quantum Logics

Quantum Logics Defined by Divisibility Conditions

On the Set-Representable Orthomodular Posets that are Point-Distinguishing

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