Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An orthomodular lattice admitting no group-valued measure

Author: Mirko Navara
Journal: Proc. Amer. Math. Soc. 122 (1994), 7-12
MSC: Primary 06C15; Secondary 03G12, 28B10
MathSciNet review: 1191871
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct a finite orthomodular lattice L such that, for each commutative group G, there is no nontrivial G-valued measure on L. This result extends a result of R. J. Greechie (Orthogonal lattices admitting no states, J. Combin. Theory Ser. A 10 (1971), 119-132), and also sheds light on recent investigations in the noncommutative measure theory.

References [Enhancements On Off] (What's this?)

  • [1] James K. Brooks and Robert S. Jewett, On finitely additive vector measures, Proc. Nat. Acad. Sci. U.S.A. 67 (1970), 1294–1298. MR 0269802
  • [2] Anna Bruna D’Andrea and Paolo De Lucia, The Brooks-Jewett theorem on an orthomodular lattice, J. Math. Anal. Appl. 154 (1991), no. 2, 507–522. MR 1088647, 10.1016/0022-247X(91)90054-4
  • [3] P. de Lucia and P. Morales, A non-commutative version of a theorem of Marczewski for submeasures, Studia Math. (to appear).
  • [4] P. de Lucia and T. Traynor, Non-commutative group-valued measures on an orthomodular poset (to appear).
  • [5] R. J. Greechie, Orthomodular lattices admitting no states, J. Combinatorial Theory Ser. A 10 (1971), 119–132. MR 0274355
  • [6] R. J. Greechie and F. R. Miller, On structures related to states on an empirical logic I. Weights on finite spaces, Technical Report 16, Dept. of Math., Kansas State Univ., Manhattan, KA, 1970.
  • [7] Stanley P. Gudder, Stochastic methods in quantum mechanics, North-Holland, New York-Oxford, 1979. North-Holland Series in Probability and Applied Mathematics. MR 543489
  • [8] Jan Hamhalter and Pavel Pták, Hilbert-space-valued states on quantum logics, Appl. Math. 37 (1992), no. 1, 51–61. MR 1152157
  • [9] Gudrun Kalmbach, Orthomodular lattices, London Mathematical Society Monographs, vol. 18, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1983. MR 716496
  • [10] A. G. Kurosh., Group theory, Nauka, Moskow, 1967. (Russian)
  • [11] Mirko Navara, Pavel Pták, and Vladimír Rogalewicz, Enlargements of quantum logics, Pacific J. Math. 135 (1988), no. 2, 361–369. MR 968618
  • [12] Mirko Navara and Vladimír Rogalewicz, Construction of orthomodular lattices with given state spaces, Demonstratio Math. 21 (1988), no. 2, 481–493. MR 981700
  • [13] Mirko Navara and Vladimír Rogalewicz, State isomorphism of orthomodular posets and hypergraphs, Proceedings of the First Winter School on Measure Theory (Liptovský Ján, 1988) Slovak Acad. Sci., Bratislava, 1988, pp. 93–98. MR 1000196
  • [14] M. Navara and G. T. Rüttimann, A characterization of 𝜎-state spaces of orthomodular lattices, Exposition. Math. 9 (1991), no. 3, 275–284. MR 1121158
  • [15] Pavel Pták, Exotic logics, Colloq. Math. 54 (1987), no. 1, 1–7. MR 928651
  • [16] P. Pták and S. Pulmannová, Orthomodular structures as quantum logics, Kluwer, Dordrecht, Boston, and London, 1991.
  • [17] Frederic W. Shultz, A characterization of state spaces of orthomodular lattices, J. Combinatorial Theory Ser. A 17 (1974), 317–328. MR 0364042
  • [18] V. S. Varadarajan, Geometry of quantum theory. Vol. I, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. The University Series in Higher Mathematics. MR 0471674
  • [19] Hans Weber, Compactness in spaces of group-valued contents, the Vitali-Hahn-Saks theorem and Nikodým’s boundedness theorem, Rocky Mountain J. Math. 16 (1986), no. 2, 253–275. MR 843053, 10.1216/RMJ-1986-16-2-253

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 06C15, 03G12, 28B10

Retrieve articles in all journals with MSC: 06C15, 03G12, 28B10

Additional Information

Keywords: Orthomodular lattice, group-valued measure, Greechie diagram
Article copyright: © Copyright 1994 American Mathematical Society