Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Mathematica Slovaca

Editor-in-Chief: Pulmannová, Sylvia


IMPACT FACTOR 2018: 0.490

CiteScore 2018: 0.47

SCImago Journal Rank (SJR) 2018: 0.279
Source Normalized Impact per Paper (SNIP) 2018: 0.627

Mathematical Citation Quotient (MCQ) 2018: 0.29

Online
ISSN
1337-2211
See all formats and pricing
More options …
Volume 65, Issue 6

Issues

Further Remarks on an Order for Quantum Observables

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

Abstract

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

References

  • [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.

Export Citation

Mathematical Institute Slovak Academy of Sciences.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Martin Bohata and Jan Hamhalter
Linear and Multilinear Algebra, 2016, Volume 64, Number 12, Page 2519

Comments (0)

Please log in or register to comment.
Log in