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.

orthomodular poset, full set of states, numerical event

06C15, 03G12, 81P16

- E. G.~Beltrametti, D.~Dorninger and M J. M{\c{a}}czy\'nski: On a cryptographical characterization of classical and nonclassical event systems. Rep.\ Math.\ Phys.\ {\mi 60} (2007), 117-123. CrossRef
- E. G.~Beltrametti and M. J.~M{\c{a}}czy\'nski: On a characterization of classical and nonclassical probabilities. J.\ Math.\ Phys.\ 32 (1991), 1280-1286. CrossRef
- E. G.~Beltrametti and M. J.~M{\c{a}}czy\'nski: On the characterization of probabilities: A~generalization of Bell's inequalities. J.\ Math.\ Phys.\ 34 (1993), 4919-4929. CrossRef
- G.~Dorfer, D.~Dorninger and H.~L\"anger: On algebras of multidimensional probabilities. Math.\ Slovaca 60 (2010), 571-582. CrossRef
- D.~Dorninger and H.~L\"anger: On a characterization of physical systems by spaces of numerical events. ARGESIM Rep.\ 35 (2009), 601-607. CrossRef
- G.~Kalmbach: Orthomodular Lattices. Academic Press, London 1983. CrossRef
- M. J.~M{\c{a}}czy\'nski and T. Traczyk: A characterization of orthomodular partially ordered sets admitting a full set of states. Bull.\ Acad.\ Polon.\ Sci.\ {\mi 21} (1973), 3-8. CrossRef
- P.~Pt\'ak: Concrete quantum logics. Internat.\ J.\ Theoret.\ Phys.\ 39 (2000), 827-837. CrossRef