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States on symmetric logics: extensions

Airat Bikchentaev and Mirko Navara
From the journal Mathematica Slovaca

Abstract

We continue the study of symmetric logics, i.e., collections of subsets generalizing Boolean algebras and closed under the symmetric difference. We contribute to several open questions. One of them is whether there is á non-Boolean symmetric logic such that all states on it are Δ-subadditive.

  1. Dedicated to Professor Anatolij Dvurecenskij

    (Communicated by Sylvia Pulmannová)

Acknowledgement

The authors thank the anonymous referees and non-anonymous colleagues for remarks that helped to improve the paper.

References

[1] Bikchentaev, A. M.: States on symmetric logics: conditional probability and independence, Lobachevskii J. Math. 30 (2009), 101–106. Search in Google Scholar

[2] Bikchentaev, A. M.—Yakushev, R. S.: States on symmetric logics: conditional probability and independence II, Internat. J. Theoret. Phys. 53 (2014), 397–408, DOI: 10.1007/s10773-013-1824-8. Search in Google Scholar

[3] De Simone, A.—Navara, M.—PTáK, P.: Extending states on finite concrete logics, Internat. J. Theoret. Phys. 46 (2007), 2046–2052. Search in Google Scholar

[4] De Simone, A.—Navara, M.—PTáK, P.: States on systems of sets that are closed under symmetric difference, Math. Nachr. 288 (2015), 1995–2000. DOI 10.1002/mana.201500029. Search in Google Scholar

[5] DiedonnÉ, J. A.: Foundations of Modern Analysis, Academic Press, New York, 1969. Search in Google Scholar

[6] Gudder, S.: Stochastic Methods in Quantum Mechanics, North-Holland, New York, 1979. Search in Google Scholar

[7] Kalmbach, G.: Orthomodular Lattices, Academic Press, London, 1983. Search in Google Scholar

[8] Matousek, M.—PTáK, P.: Symmetric difference on orthomodular lattices and Z2-valued states, Comment. Math. Univ. Carolin. 50 (2009), 535–547. Search in Google Scholar

[9] Mayet, R.—Navara, M.: Classes of logics representable as kernels of measures. In: Contributions to General Algebra 9 (G. Pilz, ed.), Teubner, Stuttgart-Wien, 1995, pp. 241–248. Search in Google Scholar

[10] MÜUller, V.: Jauch–Piron states on concrete quantum logics, Internat. J. Theoret. Phys. 32 (1993), 433–442. [11] Navara, M.: Quantum logics representable as kernels of measures, Czechoslovak Math. J. 46 (1996), 587–597. Search in Google Scholar

[11] Navara, M.: Quantum logics representable as kernels of measures, Czechoslovak Math. J. 46 (1996), 587–597. Search in Google Scholar

[12] Ovchinnikov, P. G.—Sultanbekov, F. F.: Finite concrete logics: their structure and measures on them, Internat. J. Theoret. Phys. 37 (1998), 147–153. Search in Google Scholar

[13] Ovchinnikov, P. G.: Measures on finite concrete logics, Proc. Amer. Math. Soc. 127 (1999), 1957–1966. Search in Google Scholar

[14] PTáK, P.: Some nearly Boolean orthomodular posets, Proc. Amer. Math. Soc. 126 (1998), 2039–2046. Search in Google Scholar

[15] PTáK, P.: Concrete quantum logics, Internat. J. Theoret. Phys. 39 (2000), 827–837. Search in Google Scholar

[16] Sultanbekov, F. F.: Set logics and their representations, Internat. J. Theoret. Phys. 32 (1993), 2177–2186. Search in Google Scholar

Received: 2014-2-12
Accepted: 2014-9-3
Published Online: 2016-6-5
Published in Print: 2016-4-1

© 2016 Mathematical Institute Slovak Academy of Sciences