Orthocomplemented Posets with a Symmetric Difference

Matoušek, Milan; Pták, Pavel

doi:10.1007/s11083-008-9102-8

Cited by 15 publications

(4 citation statements)

References 17 publications

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“…Due to the above mentioned properties, orthomodular posets were and are studied by numerous authors for several decades see e.g. [5,7,8,12,13]. However, up to now, orthomodular posets were treated as partial algebras where the binary operation of supremum is ensured only for orthogonal or comparable elements.…”

Section: Introductionmentioning

confidence: 99%

Variety of orthomodular posets

Chajda¹,

Kolařík²

2014

MMN

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Orthomodular posets play an important role in the so-called logical structure of a physical system as formerly pointed out by numerous authors. In particular, they play an essential role in the logic of quantum mechanics. To avoid usual problems with partial algebras, we define the so-called orthomodular directoid as an everywhere defined algebra and we show that every orthomodular poset can be converted into an orthomodular directoid and vice versa. Since orthomodular directoids are defined equationally, they form a variety having nice congruence properties.

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Section: Introductionmentioning

confidence: 99%

Variety of orthomodular posets

Chajda¹,

Kolařík²

2014

MMN

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“…So P can be represented as a collection of subsets of a set with the inclusion for ≤ and the set-complement for ⊥ , with the presence of the set symmetric difference. This has been proved in [8] by considering the two-valued △-states on P . In the "ideal" reformulation à la Boolean algebras, the ODP set representation of P follows from the existence of the orderdetermining collection of maximal Frink ideals in P .…”

Section: Resultsmentioning

confidence: 96%

“…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%

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Quantum logics close to Boolean algebras

Navara,

Pták

2021

Preprint

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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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Orthocomplemented lattices with a symmetric difference

Matoušek

2009

Algebra univers.

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Modelling an abstract version of the set-theoretic operation of symmetric difference, we first introduce the class of orthocomplemented difference lattices (ODL). We then exhibit examples of ODLs and investigate their basic properties finding, for instance, that any ODL induces an orthomodular lattice (OML) but not all OMLs can be converted to ODLs. We then analyse an appropriate version of ideals and valuations in ODLs and show that the set-representable ODLs form a variety. We finally investigate the question of constructing ODLs from Boolean algebras and obtain, as a by-product, examples of ODLs that are not set-representable but that "live" on set-representable OMLs.

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Orthocomplemented Posets with a Symmetric Difference

Cited by 15 publications

References 17 publications

Variety of orthomodular posets

Variety of orthomodular posets

Quantum logics close to Boolean algebras

Orthocomplemented lattices with a symmetric difference

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