# On the structure of numerical event spaces

Kybernetika (2010)

• Volume: 46, Issue: 6, page 971-981
• ISSN: 0023-5954

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## Abstract

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The probability $p\left(s\right)$ 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 $\left[0,1\right]$. 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.

## How to cite

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Dorfer, Gerhard, Dorninger, Dietmar W., and Länger, Helmut. "On the structure of numerical event spaces." Kybernetika 46.6 (2010): 971-981. <http://eudml.org/doc/197168>.

@article{Dorfer2010,
abstract = {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.},
author = {Dorfer, Gerhard, Dorninger, Dietmar W., Länger, Helmut},
journal = {Kybernetika},
keywords = {orthomodular poset; full set of states; numerical event; orthomodular poset; state; full set of states; concrete logic; Boolean algebra; fuzzy set},
language = {eng},
number = {6},
pages = {971-981},
publisher = {Institute of Information Theory and Automation AS CR},
title = {On the structure of numerical event spaces},
url = {http://eudml.org/doc/197168},
volume = {46},
year = {2010},
}

TY - JOUR
AU - Dorfer, Gerhard
AU - Dorninger, Dietmar W.
AU - Länger, Helmut
TI - On the structure of numerical event spaces
JO - Kybernetika
PY - 2010
PB - Institute of Information Theory and Automation AS CR
VL - 46
IS - 6
SP - 971
EP - 981
AB - 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.
LA - eng
KW - orthomodular poset; full set of states; numerical event; orthomodular poset; state; full set of states; concrete logic; Boolean algebra; fuzzy set
UR - http://eudml.org/doc/197168
ER -

## References

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1. Beltrametti, E. G., Dorninger, D., Ma̧czyński, M J., 10.1016/S0034-4877(07)80103-0, Rep. Math. Phys. 60 (2007), 117–123. (2007) Zbl1134.81307MR2355470DOI10.1016/S0034-4877(07)80103-0
2. Beltrametti, E. G., Ma̧czyński, M. J., 10.1063/1.529326, J. Math. Phys. 32 (1991), 1280–1286. (1991) MR1103482DOI10.1063/1.529326
3. Beltrametti, E. G., Ma̧czyński, M. J., On the characterization of probabilities: A generalization of Bell’s inequalities, J. Math. Phys. 34 (1993), 4919–4929. (1993) MR1243116
4. Dorfer, G., Dorninger, D., Länger, H., 10.2478/s12175-010-0032-8, Math. Slovaca 60 (2010), 571–582. (2010) Zbl1249.06023MR2728523DOI10.2478/s12175-010-0032-8
5. Dorninger, D., Länger, H., On a characterization of physical systems by spaces of numerical events, ARGESIM Rep. 35 (2009), 601–607. (2009)
6. Kalmbach, G., Orthomodular Lattices, Academic Press, London 1983. (1983) Zbl0528.06012MR0716496
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. (1973) MR0314708
8. Pták, P., 10.1023/A:1003626929648, Internat. J. Theoret. Phys. 39 (2000), 827–837. (2000) MR1792201DOI10.1023/A:1003626929648

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