Geodesic
On the structure of numerical event spaces
Kybernetika, Tome 46 (2010) no. 6, pp. 971-981

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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.
Classification : 03G12, 06C15, 81P16
Keywords: orthomodular poset; full set of states; numerical event 
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     author = {Dorfer, Gerhard and Dorninger, Dietmar and L\"anger, Helmut},
     title = {On the structure of numerical event spaces},
     journal = {Kybernetika},
     pages = {971--981},
     publisher = {mathdoc},
     volume = {46},
     number = {6},
     year = {2010},
     mrnumber = {2797421},
     zbl = {1221.06009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2010__46_6_a4/}
}
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Dorfer, Gerhard; Dorninger, Dietmar; Länger, Helmut. On the structure of numerical event spaces. Kybernetika, Tome 46 (2010) no. 6, pp. 971-981. http://geodesic.mathdoc.fr/item/KYB_2010__46_6_a4/