Filters and supports in orthoalgebras

Foulis, David J.; Greechie, Richard J.; Rüttimann, Gottfried T.

doi:10.1007/bf00678545

Cited by 142 publications

(100 citation statements)

References 13 publications

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“…Since their inception, these structures have received a good amount of attention. For a detailed account of orthoalgebras the reader should consult (Foulis et al, 1992;Wilce, 2000), and for general background on orthomodular posets and lattices the reader should consult (Kalmbach, 1983;Ptak and Pulmannová, 1991). …”

Section: Decompositions In An Honest Categorymentioning

confidence: 99%

Orthomodularity of Decompositions in a Categorical Setting

Harding

2006

Int J Theor Phys

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We provide a method to construct a type of orthomodular structure known as an orthoalgebra from the direct product decompositions of an object in a category that has finite products and whose ternary product diagrams give rise to certain pushouts. This generalizes a method to construct an orthomodular poset from the direct product decompositions of familiar mathematical structures such as non-empty sets, groups, and topological spaces, as well as a method to construct an orthomodular poset from the complementary pairs of elements of a bounded modular lattice.

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Section: Decompositions In An Honest Categorymentioning

confidence: 99%

Orthomodularity of Decompositions in a Categorical Setting

Harding

2006

Int J Theor Phys

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“…They can also be characterized as effect algebras in which, for every orthogonal pair of elements a, b, a ⊕ b = a ∨ b ( [4,5]). It is therefore obvious that, in orthomodular posets, coexistence is the same thing as the usual compatibility/commutativity.…”

Section: Remark 14ºmentioning

confidence: 99%

Spectral automorphisms in CB-effect algebras

Caragheorgheopol

2012

Mathematica Slovaca

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ABSTRACT. Spectral automorphisms have been introduced in [IVANOV, A.-CARAGHEORGHEOPOL, D.: Spectral automorphisms in quantum logics, Internat. J. Theoret. Phys. 49 (2010), 3146-3152] in an attempt to construct, in the abstract framework of orthomodular lattices, an analogue of the spectral theory in Hilbert spaces. We generalize spectral automorphisms to the framework of effect algebras with compression bases and study their properties. Characterizations of spectral automorphisms as well as necessary conditions for an automorphism to be spectral are given. An example of a spectral automorphism on the standard effect algebra of a finite-dimensional Hilbert space is discussed and the consequences of spectrality of an automorphism for the unitary Hilbert space operator that generates it are shown.The last section is devoted to spectral families of automorphisms and their properties, culminating with the formulation and proof of a Stone type theorem (in the sense of Stone's theorem on strongly continuous one-parameter unitary groups -see, e.g. Orthomodular lattices and orthomodular posets have been considered for a long time to be the appropriate mathematical structures for studying the logic of quantum mechanics [13,14]. To allow the study of unsharp measurements or observations, represented by quantum effects, the algebraic structure of effect algebra is currently used (see, e.g., [2,4]). The so-called compressions were introduced by Gudder [7] in the framework of effect algebras as an abstraction of 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 81P10; Secondary 81Q10. K e y w o r d s: spectral automorphism, effect algebra, compression base, CB-effect algebra, spectral theory, effect algebra automorphism. The author gratefully acknowledges the scholarship of the Czech Technical University in Prague.

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“…Let us remark that an orthomodular poset is usually defined as a bounded partially ordered set with an orthocomplementation in which the orthomodular law is valid. (See [3,4]) Definition 1.4 Let E be an effect algebra. The isotropic index i(a) of an element a ∈ E is sup{n ∈ N : na is defined}, where na = n i=1 a is the sum of n copies of a.…”

Section: Definition 13mentioning

confidence: 99%

Common Generalizations of Orthocomplete and Lattice Effect Algebras

Tkadlec

2009

Int J Theor Phys

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Connections between the weak orthocompleteness and the maximality property in effect algebras are presented. It is proved that an orthomodular poset with the maximality property is disjunctive. A characterization of Archimedean weakly orthocomplete effect algebras is given.Keywords Effect algebra · weakly orthocomplete · maximality property · disjunctive · Archimedean · separable Ovchinnikov [7] introduced weakly orthocomplete orthomodular posets (he called them alternative) as a common generalization of orthocomplete orthomodular posets and orthomodular lattices and showed that they are disjunctive. Weak orthocompleteness is useful in the study of orthoatomisticity and disjunctivity might be used to characterize atomisticity [7,11]. Weak orthocompleteness was generalized by De Simone and Navara [1].Tkadlec [8] introduced the class of orthomodular posets with the maximality property as another common generalization of orthocomplete orthomodular posets and orthomodular lattices. He showed various consequences of this property and generalize it [8,9,10,12].We show that these two notions are incomparable, that maximality property also implies disjunctivity in orthomodular posets and present a characterization of Archimedean weakly orthocomplete effect algebras. We show also some other relations within the class of effect algebras.

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Filters and supports in orthoalgebras

Cited by 142 publications

References 13 publications

Orthomodularity of Decompositions in a Categorical Setting

Orthomodularity of Decompositions in a Categorical Setting

Spectral automorphisms in CB-effect algebras

Common Generalizations of Orthocomplete and Lattice Effect Algebras

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