Quantum Logics that are Symmetric-difference-closed

Burešová, Dominika; Pták, Pavel

doi:10.1007/s10773-021-04950-6

Cited by 4 publications

(4 citation statements)

References 14 publications

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“…The statement of Theorem 2.5(iii) could be instrumental in the study of the orthomodular posets of SOMP that have a symmetric difference (see e.g. [2,4,10]). In particular, it is worth noticing that the examples of conceptually important orthomodular posets with the property that x is compatible with y exactly when x ∨ y exists (see [13]) could be constructed point-distinguishing.…”

Section: Obviously ( Pmentioning

confidence: 99%

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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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Section: Obviously ( Pmentioning

confidence: 99%

“…They are called quantum logics. Among those orthomodular posets, a special conceptual position is taken by the set-representable ones (see [2,3,4,5,6,7,8,14] etc.). In this note, we shall exclusively deal with set-representable orthomodular posets.…”

Section: Introductionmentioning

confidence: 99%

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

Burešová,

Pták

2023

Int J Theor Phys

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“…The investigation of symmetric-difference-closed orthomodular posets and lattices has recently brought a substantial contribution to the theory of orthocomplemented structures (see [1][2][3][4][5][6][7][8][9]). In this note, we ask on the presence of the "Greechie phenomenon" [10] in ODLs: Is there a stateless ODL?…”

Section: Notionsmentioning

confidence: 99%

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

Voráček

Pták

2022

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This paper carries on the investigation of the orthomodular lattices that are endowed with a symmetric difference. Let us call them ODLs. Note that the ODLs may have a certain bearing on "quantum logics" -the ODLs are close to Boolean algebras though they capture the phenomenon of non-compatibility. The initial question in studying the state space of the ODLs is whether the state space can be poor. This question is of a purely combinatorial nature. In this note, we exhibit a finite ODL whose state space is empty (respectively, whose state space is a singleton).

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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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Quantum Logics that are Symmetric-difference-closed

Cited by 4 publications

References 14 publications

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

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

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

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

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