States On Orthocomplemented Difference Posets (Extensions)

Hroch, Michal; Pták, Pavel

doi:10.1007/s11005-016-0862-6

Cited by 8 publications

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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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Section: Notionsmentioning

confidence: 99%

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

Voráček

Pták

2022

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“…So P is orthomodular and hence the ODPs are a kind of enriched quantum logics. (We shall study certain algebraic properties of ODPs; the state properties have been investigated in [6] and [8]).…”

Section: Notations Used Throughout the Papermentioning

confidence: 99%

Quantum logics close to Boolean algebras

Navara,

Pták

2021

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We consider orthomodular posets endowed with a symmetric difference. We call them ODPs. Expressed in the quantum logic language, we consider quantum logics with an XOR-type connective. We study three classes of "almost Boolean" ODPs, two of them defined by requiring rather specific behaviour of infima and the third by a Boolean-like behaviour of Frink ideals. We establish a (rather surprising) inclusion between the three classes, shadding thus light on their intrinsic properties. (More details can be found in the Introduction that follows.) Let us only note that the orthomodular posets pursued here, though close to Boolean algebras (i.e., close to standard quantum logics), still have a potential for an arbitrarily high degree of non-compatibility and hence they may enrich the studies of mathematical foundations of quantum mechanics.

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“…Another fact worth observing is that each set-representable OMP can be OMP embedded into an OMP that belongs to R. To show that, if we "replace", in a set-representable OMP P , each point of the underlying set of P by an infinite set we thus obtain an OMP, Q, isomorphic to P . If one adds to Q all finite and co-finite sets, then this collection generates an OMP that belongs to R (see also [5]). As a result, the class R is in a sense as "big" as the class of all set-representable OMPs.…”

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On Frink Ideals in Orthomodular Posets

Navara

Pták

2020

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Let S denote the class of orthomodular posets in which all maximal Frink ideals are selective. Let R (resp. T ) be the class of orthomodular posets defined by the validity of the following implications:In this note we prove the following slightly surprising result: R ⊂ S ⊂ T . Since orthomodular posets are often understood as quantum logics, the result might have certain bearing on quantum axiomatics. Notions and ResultsWe study three classes of orthomodular posets introduced in the abstract. Though the classes R and T have naturally appeared in the investigation of orthomodular posets ([5-8, 11]), and they may have certain interpretation in theoretical physics as quantum logics (for instance, if P ∈ R and a, b ∈ P , then a, b are compatible exactly when a ∧ b exists in P , see [8]), the link with purely order-theoretic notion of Frink ideal may appear unexpected. Prior to formulating our results, let us recall the definitions as we shall use them in the sequel. We shall exclusively deal with orthomodular posets (OMPs, see [10]). By an OMP we mean a 5-tuple (P , ≤, , 0, 1) such that the following conditions are fulfilled:(1) (P , ≤) is a partially ordered set with a least element, 0, and a greatest element, 1;

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States On Orthocomplemented Difference Posets (Extensions)

Cited by 8 publications

References 9 publications

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

Quantum logics close to Boolean algebras

On Frink Ideals in Orthomodular Posets

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