Kybernetika 47 no. 4, 541-559, 2011

Observables on sigma-MV algebras and sigma-lattice effect algebras

Anna Jenčová, Silvia Pulmannová and Elena Vinceková


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.


state, observable, MV algebra, lattice effect algebra, Markov kernel, weak Markov kernel, smearing, generalized observable


81P10, 81P15, 03G12


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