| Title:
|
On the structure of numerical event spaces (English) |
| Author:
|
Dorfer, Gerhard |
| Author:
|
Dorninger, Dietmar |
| Author:
|
Länger, Helmut |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 |
| Volume:
|
46 |
| Issue:
|
6 |
| Year:
|
2010 |
| Pages:
|
971-981 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The probability $p(s)$ of the occurrence of an event pertaining to a physical system which is observed in different states $s$ determines a function $p$ from the set $S$ of states of the system to $[0,1]$. The function $p$ is called a numerical event or multidimensional probability. When appropriately structured, sets $P$ of numerical events form so-called algebras of $S$-probabilities. Their main feature is that they are orthomodular partially ordered sets of functions $p$ with an inherent full set of states. A classical physical system can be characterized by the fact that the corresponding algebra $P$ of $S$-probabilities is a Boolean lattice. We give necessary and sufficient conditions for systems of numerical events to be a lattice and characterize those systems which are Boolean. Assuming that only a finite number of measurements is available our focus is on finite algebras of $S$-probabilties. (English) |
| Keyword:
|
orthomodular poset |
| Keyword:
|
full set of states |
| Keyword:
|
numerical event |
| MSC:
|
03G12 |
| MSC:
|
06C15 |
| MSC:
|
81P16 |
| idZBL:
|
Zbl 1221.06009 |
| idMR:
|
MR2797421 |
| . |
| Date available:
|
2011-04-12T12:43:42Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141460 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[4] Dorfer, G., Dorninger, D., Länger, H.: On algebras of multidimensional probabilities.Math. Slovaca 60 (2010), 571–582. Zbl 1249.06023, MR 2728523, 10.2478/s12175-010-0032-8 |
| Reference:
|
[5] Dorninger, D., Länger, H.: On a characterization of physical systems by spaces of numerical events.ARGESIM Rep. 35 (2009), 601–607. |
| Reference:
|
[6] Kalmbach, G.: Orthomodular Lattices.Academic Press, London 1983. Zbl 0528.06012, MR 0716496 |
| Reference:
|
[7] Ma̧czyński, M. J., Traczyk, T.: A characterization of orthomodular partially ordered sets admitting a full set of states.Bull. Acad. Polon. Sci. 21 (1973), 3–8. MR 0314708 |
| Reference:
|
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| . |
