Group-valued measures on coarse-grained quantum logics

Anna de Simone; Pavel Pták

Czechoslovak Mathematical Journal (2007)

  • Volume: 57, Issue: 2, page 737-746
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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de Simone, Anna, and Pták, Pavel. "Group-valued measures on coarse-grained quantum logics." Czechoslovak Mathematical Journal 57.2 (2007): 737-746. <http://eudml.org/doc/31159>.

@article{deSimone2007,
abstract = {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.},
author = {de Simone, Anna, Pták, Pavel},
journal = {Czechoslovak Mathematical Journal},
keywords = {coarse-grained quantum logic; group-valued measure; measure extension; coarse-grained quantum logic; group-valued measure; measure extension},
language = {eng},
number = {2},
pages = {737-746},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Group-valued measures on coarse-grained quantum logics},
url = {http://eudml.org/doc/31159},
volume = {57},
year = {2007},
}

TY - JOUR
AU - de Simone, Anna
AU - Pták, Pavel
TI - Group-valued measures on coarse-grained quantum logics
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 2
SP - 737
EP - 746
AB - 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.
LA - eng
KW - coarse-grained quantum logic; group-valued measure; measure extension; coarse-grained quantum logic; group-valued measure; measure extension
UR - http://eudml.org/doc/31159
ER -

References

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  9. Measures on the Gudder-Marchand logics, Constructive Theory of Functions and Functional Analysis 8 (1992), 95–98. (Russian) (1992) MR1231108
  10. 10.1090/S0002-9939-98-04403-7, Proc. Amer. Math. Soc. 126 (1998), 2039–2046. (1998) MR1452822DOI10.1090/S0002-9939-98-04403-7
  11. 10.1023/A:1003626929648, Internat. J. Theoret. Phys. 39 (2000), 827–837. (2000) MR1792201DOI10.1023/A:1003626929648
  12. Orthomodular Structures as Quantum Logics, Kluwer, Dordrecht/Boston/London, 1991. (1991) MR1176314

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