Orthomodular lattices with state-separated noncompatible pairs

Mayet, R.; Pták, Pavel

doi:10.1023/a:1022479104160

Cited by 5 publications

(3 citation statements)

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“…It could be noted that Theorem 2.5(ii) may shed light on the algebraic theory of the subclass of SOMP consisting of lattices-this subclass forms a variety (see [6,11,12]; obviously, the point-distinguishing lattices of SOMP are easier to deal with). 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.…”

Section: Obviously ( Pmentioning

confidence: 99%

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%

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

Burešová,

Pták

2023

Int J Theor Phys

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“…In the rest of this section we shall be proving that the class SRODL is a variety. It should be noted that the central strategic line of the investigation of setrepresentable OMLs as used in [11,19,20,21] was instrumental in places. However, the presence of the operation △ required to invent some new techniques.…”

Section: Set-representable Odlsmentioning

confidence: 99%

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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“…Hence the class SRODP is rather large and algebraically "stylish". It should be noted that in showing that SRODP is a quasivariety the investigation of the set-representation of orthomodular posets was instrumental (see [18] and [19]). However, the presence of the extra operation required here a somewhat different reasoning in places.…”

Section: Set-representable Odps Form a Quasivarietymentioning

confidence: 99%

Orthocomplemented Posets with a Symmetric Difference

Matoušek

Pták

2009

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We endow orthocomplemented posets with a binary operation-an abstract symmetric difference of sets-and we study algebraic properties of this class, ODP. Denoting its elements by ODP, we first investigate on the features related to compatibility in ODPs. We find, among others, that any ODP is orthomodular. This explicitly links ODP with the theory of quantum logics. By analogy with Boolean algebras, we then ask if (when) an ODP is set representable. Though we find that general ODPs do not have to be set representable, many natural ODPs are shown to be. We characterize the set-representable ODPs in terms of two valued morphisms and prove that they form a quasivariety. This quasivariety contains the class of pseudocomplemented ODPs as we show afterwards. At the end we ask whether any orthomodular poset can be converted or, more generally, enlarged to an ODP. By countre-examples we answer these questions to the negative.

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Orthomodular lattices with state-separated noncompatible pairs

Cited by 5 publications

References 15 publications

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

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

Orthocomplemented lattices with a symmetric difference

Orthocomplemented Posets with a Symmetric Difference

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