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coarse-grained quantum logic; group-valued measure; measure extension
In 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 () 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.
[1] M. R. Darnel: Theory of Lattice-Ordered Groups. Dekker, New York, 1995. MR 1304052 | Zbl 0810.06016
[2] A. De Simone, M. Navara and P. Pták: Extensions of states on concrete finite logics. (to appear).
[3] S. Gudder and J. P. Marchand: A coarse-grained measure theory. Bull. Polish Acad. Sci. Math. 28 (1980), 557–564. MR 0628642
[4] S. Gudder: Stochastic Methods in Quantum Mechanics. North Holland, 1979. MR 0543489 | Zbl 0439.46047
[5] S. Gudder: Quantum probability spaces. Proc. Amer. Math. Soc. 21 (1969), 286–302. MR 0243793 | Zbl 0183.28703
[6] S. Gudder: An extension of classical measure theory. SIAM 26 (1984), 71–89. DOI 10.1137/1026002 | MR 0735076 | Zbl 0559.28003
[7] P. de Lucia and P. Pták: Quantum logics with classically determined states. Colloq. Math. 80 (1999), 147–154. MR 1684578
[8] M. Navara and P. Pták: Almost Boolean orthomodular posets. J. Pure Appl. Algebra 60 (1989), 105–111. DOI 10.1016/0022-4049(89)90108-4 | MR 1014608
[9] P. Ovtchinikoff: Measures on the Gudder-Marchand logics. Constructive Theory of Functions and Functional Analysis 8 (1992), 95–98. (Russian) MR 1231108
[10] P. Pták: Some nearly Boolean orthomodular posets. Proc. Amer. Math. Soc. 126 (1998), 2039–2046. DOI 10.1090/S0002-9939-98-04403-7 | MR 1452822
[11] P. Pták: Concrete quantum logics. Internat. J. Theoret. Phys. 39 (2000), 827–837. DOI 10.1023/A:1003626929648 | MR 1792201
[12] P. Pták and S. Pulmannová: Orthomodular Structures as Quantum Logics. Kluwer, Dordrecht/Boston/London, 1991. MR 1176314
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