# On the structure of numerical event spaces

Gerhard Dorfer; Dietmar W. Dorninger; Helmut Länger

Kybernetika (2010)

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

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topDorfer, 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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- 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
- Dorninger, D., Länger, H., On a characterization of physical systems by spaces of numerical events, ARGESIM Rep. 35 (2009), 601–607. (2009)
- Kalmbach, G., Orthomodular Lattices, Academic Press, London 1983. (1983) Zbl0528.06012MR0716496
- 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
- Pták, P., 10.1023/A:1003626929648, Internat. J. Theoret. Phys. 39 (2000), 827–837. (2000) MR1792201DOI10.1023/A:1003626929648

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