Czechoslovak Mathematical Journal, Vol. 57, No. 2, pp. 737-746, 2007

Group-valued measures on coarse-grained
quantum logics

Anna De Simone, Pavel Ptak

Anna De Simone, Dipartimento di Matematica, e Statistica, Universita degli Studi, "Federico II" di Napoli, 80126 Napoli, Italy, e-mail:; Pavel Ptak, Czech Technical University, Faculty of Electrical Eng., Department of Mathematics, 166 27 Prague 6, Czech Republic, e-mail:

Abstract: In \cite{GudMar} it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later (\cite{Ovchi}) this result was re-proved (and further improved on) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof technique is entirely different from that of the preceding papers. In addition, we provide a new combinatorial argument for describing all atoms of cyclic coarse-grained quantum logics.

Keywords: coarse-grained quantum logic, group-valued measure, measure extension

Classification (MSC 2000): 06C15, 81P10, 28A99, 28A55

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