Previous |  Up |  Next


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:
Reference: [1] Beltrametti, E. G., Dorninger, D., Ma̧czyński, M J.: On a cryptographical characterization of classical and nonclassical event systems.Rep. Math. Phys. 60 (2007), 117–123. Zbl 1134.81307, MR 2355470, 10.1016/S0034-4877(07)80103-0
Reference: [2] Beltrametti, E. G., Ma̧czyński, M. J.: On a characterization of classical and nonclassical probabilities.J. Math. Phys. 32 (1991), 1280–1286. MR 1103482, 10.1063/1.529326
Reference: [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. MR 1243116
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: [8] Pták, P.: Concrete quantum logics.Internat. J. Theoret. Phys. 39 (2000), 827–837. MR 1792201, 10.1023/A:1003626929648


Files Size Format View
Kybernetika_46-2010-6_5.pdf 253.0Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo