Geodesic
Observables on $\sigma $-MV algebras and $\sigma $-lattice effect algebras
Kybernetika, Tome 47 (2011) no. 4, pp. 541-559

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Effect algebras were introduced as abstract models of the set of quantum effects which represent sharp and unsharp properties of physical systems and play a basic role in the foundations of quantum mechanics. In the present paper, observables on lattice ordered $\sigma$-effect algebras and their “smearings” with respect to (weak) Markov kernels are studied. It is shown that the range of any observable is contained in a block, which is a $\sigma$-MV algebra, and every observable is defined by a smearing of a sharp observable, which is obtained from generalized Loomis-Sikorski theorem for $\sigma$-MV algebras. Generalized observables with the range in the set of sharp real observables are studied and it is shown that they contain all smearings of observables.
Classification : 03G12, 81P10, 81P15
Keywords: lattice effect algebra; MV algebra; observable; state; Markov kernel; weak Markov kernel; smearing; generalized observable 
@article{KYB_2011__47_4_a3,
     author = {Jen\v{c}ov\'a, Anna and Pulmannov\'a, Silvia and Vincekov\'a, Elena},
     title = {Observables on $\sigma ${-MV} algebras and $\sigma $-lattice effect algebras},
     journal = {Kybernetika},
     pages = {541--559},
     publisher = {mathdoc},
     volume = {47},
     number = {4},
     year = {2011},
     mrnumber = {2884860},
     zbl = {1237.81008},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KYB_2011__47_4_a3/}
}
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Jenčová, Anna; Pulmannová, Silvia; Vinceková, Elena. Observables on $\sigma $-MV algebras and $\sigma $-lattice effect algebras. Kybernetika, Tome 47 (2011) no. 4, pp. 541-559. http://geodesic.mathdoc.fr/item/KYB_2011__47_4_a3/