Geodesic
Observables on $\sigma $-MV algebras and $\sigma $-lattice effect algebras
Kybernetika, Tome 47 (2011) no. 4, pp. 541-559 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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 
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     title = {Observables on $\sigma ${-MV} algebras and $\sigma $-lattice effect algebras},
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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/

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