Λ Note on Distributivity in Orthoposets

Tkadlec, Josef

doi:10.1515/dema-1991-1-233

Cited by 8 publications

(10 citation statements)

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“…An orthoposet is called orthocomplete if it is a-orthocomplete for every cardinal number a. The following theorem is a generalization of various results from [3,8,5,4]. Theorem 3.6.…”

Section: Definition 31 Let a Be A Cardinal Number An Orthoposet P mentioning

confidence: 96%

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Boolean orthoposets---concreteness and orthocompleteness

Tkadlec¹

1994

Math. Bohem.

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“…An orthoposet is called orthocomplete if it is a-orthocomplete for every cardinal number a. The following theorem is a generalization of various results from [3,8,5,4]. Theorem 3.6.…”

Section: Definition 31 Let a Be A Cardinal Number An Orthoposet P mentioning

confidence: 96%

“…We will use Theorem 2.6. It should be noted that an alternative proof (using some distributivity property of Boolean orthoposets) of Corollary 3.4 and Proposition 3.5 is presented in [8].…”

Section: Orthocompleteness In Boolean Orthoposetsmentioning

confidence: 99%

Boolean orthoposets---concreteness and orthocompleteness

Tkadlec¹

1994

Math. Bohem.

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“…For orthoposets (which specific sets of S-probabilities in general are not), this property is known to be Boolean (cf., e.g. Tkadlec (1991); in connection with orthomodular posets see i.a. Maczyński and Traczyk (1973) and Navara and Pták (1989)).…”

Section: (): V-volmentioning

confidence: 99%

“…By Proposition 2.2 (i), specific sets of varying S-probabilities are orthoposets and hence, Boolean GFEs are Boolean orthoposets. As mentioned in Tkadlec (1991), Boolean orthoposets are concrete logics due to a proof by Navara and Pták about Boolean orthomodular posets which does not make use of orthomodularity (cf. Navara and Pták (1989)).…”

Section: Proposition 22 a Specific Set P Of Varying S-probabilities Has The Following Propertiesmentioning

confidence: 99%

“…p ∨ q and p ∧q exist for all p, q ∈ P), then we also have a Boolean algebra (cf. Tkadlec (1991)). That P is lattice-ordered can be characterized by a simple criterion: According to Theorem 2.4, P is a concrete logic, and as shown in Dorninger and Länger (2016), a structured set of S-probabilities P which is a concrete logic is a lattice if and only if for all p, q ∈ P max( p, q) ∈ P (the maximum of the functions considered pointwise).…”

Section: Theorem 33 the Class Of Structured Sets Of S-probabilities Is A Proper Subclass Of The Class Of ∨-Specific Sets Of Sprobabilitiementioning

confidence: 99%

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On Boolean posets of numerical events

Dorninger

Länger

2021

Adv. in Comp. Int.

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With many physical processes in which quantum mechanical phenomena can occur, it is essential to take into account a decision mechanism based on measurement data. This can be achieved by means of so-called numerical events, which are specified as follows: Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state $$s\in S$$ s ∈ S . A function $$p:S\rightarrow [0,1]$$ p : S → [ 0 , 1 ] is called a numerical event or alternatively, an S-probability. If a set P of S-probabilities is ordered by the order of real functions, it becomes a poset which can be considered as a quantum logic. In case the logic P is a Boolean algebra, this will indicate that the underlying physical system is a classical one. The goal of this paper is to study sets of S-probabilities which are not far from being Boolean algebras by means of the addition and comparison of functions that occur in these sets. In particular, certain classes of so-called Boolean posets of S-probabilities are characterized and related to each other and descriptions based on sets of states are derived.

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