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Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia

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Volume 65, Issue 6


Further Remarks on an Order for Quantum Observables

Jānis Cīrulis
Published Online: 2016-02-09 | DOI: https://doi.org/10.1515/ms-2015-0109


S. Gudder and, later, S. Pulmanová and E. Vinceková, have studied in two recent papers a certain ordering of bounded self-adjoint operators on a Hilbert space. We present some further results on this ordering and show that some structure theorems of the ordered set of operators can be obtained in a more abstract setting of posets having the upper bound property and equipped with a certain orthogonality relation.

Keywords: generalized orthoalgebra; generalized orthoposet; nearsemilattice; nearlattice; orthogonality; skew meet; quasi-orthomodularity; bounded self-adjoint operator


  • [1] ANTEZANA, J.-CANO, C. et al: A note on the star order in Hilbert spaces, Linear Multilinear Algebra 58 (2010), 1037-1051.CrossrefWeb of ScienceGoogle Scholar

  • [2] CHAJDA, I.-KOLAŘĺK, M.: Nearlattices, Discrete Math. 308 (2008), 4906-4913.CrossrefWeb of ScienceGoogle Scholar

  • [3] CĪRULIS, J.: Subtractive nearsemilattices, Proc. Latv. Acad. Sci. Sect. B Nat. Exact Appl. Sci. 52 (1998), 228-233.Google Scholar

  • [4] CĪRULIS, J.: Knowledge representation in extended Pawlak’s information systems: algebraic aspects. In: FOIKS 2002 (T. Eiter, K.-D. Schewe, eds.). Lecture Notes in Comput. Sci. 2284, Springer-Verlag, Berlin, 2002, pp. 250-267.Google Scholar

  • [5] CĪRULIS, J.: Knowledge representation systems and skew nearlattices. In: Contributions to General Algebra 14 (I. Chajda, et al, eds.), Verlag Johannes Heyn, Klagenfurt, 2004, pp. 43-51.Google Scholar

  • [6] CĪRULIS, J: Skew nearlattices: some structure and representation theorems. In: Contributions to General Algebra 19 (I. Chajda, et al, eds.), Verlag Johannes Heyn, Klagenfurt, 2010, pp. 33-44.Google Scholar

  • [7] CĪRULIS, J.: Subtraction-like operations in nearsemilattices, Demonstratio Math. 43 (2010), 725-738.Google Scholar

  • [8] CORNISH, W. H.: Conversion of nearlattices into implicative BCK-algebras, Math. Semin. Notes Kobe Univ. 10 (1982), 1-8.Google Scholar

  • [9] CORNISH, W. H.-NOOR, A. S. A.: Standard elements in a nearlattice, Bull. Aust. Math. Soc. 26 (1982), 185-213.CrossrefGoogle Scholar

  • [10] DRAZIN, M. F.: Natural structures on semigroups with involution, Bull. Amer. Math. Soc. 84 (1978), 139-141.CrossrefGoogle Scholar

  • [11] DVUREČENSKIJ, A.-PULMANNOVÁ, S.: New Trends in Quantum Structures, Kluwer Acad. Publ./Ister Sci., Dordrecht/Bratislava, 2000.Google Scholar

  • [12] DU, H.-DOU, Y.: A spectral representation of the infimum of selfadjoint operators in the logic order, Acta Math. Sinica (Chin. Ser.) 52 (2009), 1141-1146 (Chinese).Google Scholar

  • [13] GUDDER, S: An order for quantum observables, Math. Slovaca 56 (2006), 573-589.Google Scholar

  • [14] JANOWITZ, M.F.: A note on generalized orthomodular lattices, J. Natur. Sci. Math. 8 (1968), 89-94.Google Scholar

  • [15] LIU,W.-WU, J.: A representation theorem of infimum of bounded observables, J.Math. Phys. 49 (2008), Article No. 073521, 5 pp.Google Scholar

  • [16] LIU, W.-WU, J.: A supremumum of bounded quantum observables, J. Math. Phys. 50 (2009), Article No. 083513, 4 pp.Google Scholar

  • [17] MAYET-IPPOLITO, A.: Generalized orthomodular posets, Demonstratio Math. 24 (1991), 263-274.Google Scholar

  • [18] NOOR, A. S. A.-CORNISH,W. H.: Multipliers on a nearlattice, Comment. Math. Univ. Carolin. 27 (1986), 815-827.Google Scholar

  • [19] PULMANNOVÁ, S.-VINCEKOVÁ, E.: Remarks on the order for quantum observables, Math. Slovaca 57 (2007), 589-600.CrossrefWeb of ScienceGoogle Scholar

  • [20] SHEN, J,-WU, J.: Spectral representation of infimum of bounded quantum observables, J. Math. Phys. 50 (2009), Article No. 1135014, 4 pp.Web of ScienceGoogle Scholar

  • [21] XU, X.-DU, H.-FANG, X.: An explicit expression of supremum of bounded quantum observables, J. Math. Phys. 50 (2009), Article No. 033502, 9 pp. Web of ScienceGoogle Scholar

About the article

Received: 2012-06-01

Accepted: 2013-04-16

Published Online: 2016-02-09

Published in Print: 2015-12-01

Citation Information: Mathematica Slovaca, Volume 65, Issue 6, Pages 1609–1626, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2015-0109.

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