Previous |  Up |  Next


orthomodular poset; full set of states; numerical event
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.
[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. DOI 10.1016/S0034-4877(07)80103-0 | MR 2355470 | Zbl 1134.81307
[2] Beltrametti, E. G., Ma̧czyński, M. J.: On a characterization of classical and nonclassical probabilities. J. Math. Phys. 32 (1991), 1280–1286. DOI 10.1063/1.529326 | MR 1103482
[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
[4] Dorfer, G., Dorninger, D., Länger, H.: On algebras of multidimensional probabilities. Math. Slovaca 60 (2010), 571–582. DOI 10.2478/s12175-010-0032-8 | MR 2728523 | Zbl 1249.06023
[5] Dorninger, D., Länger, H.: On a characterization of physical systems by spaces of numerical events. ARGESIM Rep. 35 (2009), 601–607.
[6] Kalmbach, G.: Orthomodular Lattices. Academic Press, London 1983. MR 0716496 | Zbl 0528.06012
[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
[8] Pták, P.: Concrete quantum logics. Internat. J. Theoret. Phys. 39 (2000), 827–837. DOI 10.1023/A:1003626929648 | MR 1792201
Partner of
EuDML logo