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: | [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/ |