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Content available remote

Quantum logics with classically determined states

100%
de Lucia P., Pták P.
Colloquium Mathematicum
|
1999
|
tom 80
|
nr 1
147-154
2
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On the set representation of an orthomodular poset

100%
Harding J., Pták P.
Colloquium Mathematicum
|
2001
|
tom 89
|
nr 2
233-240
EN
Let P be an orthomodular poset and let B be a Boolean subalgebra of P. A mapping s:P → ⟨0,1⟩ is said to be a centrally additive B-state if it is order preserving, satisfies s(a') = 1 - s(a), is additive on couples that contain a central element, and restricts to a state on B. It is shown that, for any Boolean subalgebra B of P, P has an abundance of two-valued centrally additive B-states. This answers positively a question raised in [13, Open question, p. 13]. As a consequence one obtains a somewhat better set representation of orthomodular posets and a better extension theorem than in [2, 12, 13]. Further improvement in the Boolean vein is hardly possible as the concluding example shows.
3
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Extending Coarse-Grained Measures

100%
De Simone A., Pták P.
Bulletin of the Polish Academy of Sciences. Mathematics
|
2006
|
tom 54
|
nr 1
1-11
EN
In [4] it is proved that a measure on a finite coarse-grained space extends, as a signed measure, over the entire power algebra. In [7] this result is reproved and further improved. Both the articles [4] and [7] use the proof techniques of linear spaces (i.e. they use multiplication by real scalars). In this note we show that all the results cited above can be relatively easily obtained by the Horn-Tarski extension technique in a purely combinatorial manner. We also characterize the pure measures and settle the dimension of the normalized-measure space. We then comment on a consequence of the results for circulant matrices. Finally, we take up the case of circle coarse-grained space and also establish a measure-extension result.
4
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Exotic logics

35%
Pták P.
Colloquium Mathematicum
|
1987
|
tom 54
|
nr 1
1-7
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