Skip to main content
×
Home
    • Aa
    • Aa
  • Access
  • Cited by 4
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Avallone, Anna and Vitolo, Paolo 2016. Pseudo-D-lattices and separating points of measures. Fuzzy Sets and Systems, Vol. 289, p. 43.


    Avallone, Anna Barbieri, Giuseppina Vitolo, Paolo and Weber, Hans 2015. Decomposition of Pseudo-effect Algebras and the Hammer–Sobczyk Theorem. Order,


    Avallone, Anna Barbieri, Giuseppina and Vitolo, Paolo 2013. Pseudo-D-lattices and Lyapunov measures. Rendiconti del Circolo Matematico di Palermo, Vol. 62, Issue. 2, p. 301.


    Lu, Xian Shang, Yun and Lu, RuQian 2012. A direct product decomposition of QMV algebras. Science China Mathematics, Vol. 55, Issue. 4, p. 841.


    ×
  • Currently known as: Journal of the Australian Mathematical Society Title history
    Journal of the Australian Mathematical Society, Volume 89, Issue 3
  • December 2010, pp. 335-358

TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA

  • DAVID J. FOULIS (a1), SYLVIA PULMANNOVÁ (a2) and ELENA VINCEKOVÁ (a3)
  • DOI: http://dx.doi.org/10.1017/S1446788711001042
  • Published online: 28 February 2011
Abstract
Abstract

Effect algebras, which generalize the lattice of projections in a von Neumann algebra, serve as a basis for the study of unsharp observables in quantum mechanics. The direct decomposition of a von Neumann algebra into types I, II, and III is reflected by a corresponding decomposition of its lattice of projections, and vice versa. More generally, in a centrally orthocomplete effect algebra, the so-called type-determining sets induce direct decompositions into various types. In this paper, we extend the theory of type decomposition to a (possibly) noncommutative version of an effect algebra called a pseudoeffect algebra. It has been argued that pseudoeffect algebras constitute a natural structure for the study of noncommuting unsharp or fuzzy observables. We develop the basic theory of centrally orthocomplete pseudoeffect algebras, generalize the notion of a type-determining set to pseudoeffect algebras, and show how type-determining sets induce direct decompositions of centrally orthocomplete pseudoeffect algebras.

    • Send article to Kindle

      To send this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA
      Your Kindle email address
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about sending content to Dropbox.

      TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about sending content to Google Drive.

      TYPE DECOMPOSITION OF A PSEUDOEFFECT ALGEBRA
      Available formats
      ×
Copyright
Corresponding author
For correspondence; e-mail: pulmann@mat.savba.sk
Footnotes
Hide All

The second and third authors were supported by the Research and Development Support Agency under contract no. LPP-0199-07; grant VEGA 2/0032/09, Center of Excellence SAS–Quantum Technologies; and ERDF OP R & D Project CE QUTE ITMS 26240120009 and meta-QUTE ITMS 26240120022.

Footnotes
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]L. Beran , Orthomodular Lattices, An Algebraic Approach, Mathematics and its Applications, 18 (D. Reidel, Dordrecht, 1985).

[2]J. C. Carrega , G. Chevalier and R. Mayet , ‘Direct decompositions of orthomodular lattices’, Algebra Universalis 27 (1990), 480496.

[3]C. C. Chang , ‘Algebraic analysis of many-valued logic’, Trans. Amer. Math. Soc. 88 (1958), 467490.

[6]A. Dvurečenskij and S. Pulmannová , New Trends in Quantum Structures (Kluwer, Dordrecht, 2000).

[8]A. Dvurečenskij and T. Vetterlein , ‘Pseudoeffect algebras. II. Group representations’, Internat. J. Theoret. Phys. 40 (2001), 703726.

[9]A. Dvurečenskij and T. Vetterlein , ‘Non-commutative algebras and quantum structures’, Internat. J. Theoret. Phys. 43(7/8) (2004), 15991612.

[10]D. J. Foulis and M. K. Bennett , ‘Effect algebras and unsharp quantum logics’, Found. Phys. 24(10) (1994), 13311352.

[13]D. J. Foulis and S. Pulmannová , ‘Type decomposition of an effect algebra’, Found. Phys. 40 (2010), 15431565.

[16]R. J. Greechie , D. J. Foulis and S. Pulmannová , ‘The center of an effect algebra’, Order 12 (1995), 91106.

[18]G. Kalmbach , Measures and Hilbert Lattices (World Scientific, Singapore, 1986).

[19]I. Kaplansky , ‘Projections in Banach algebras’, Ann. of Math. (2) 53(2) (1951), 235249.

[24]J. Rachůnek , ‘A non-commutative generalization of MV algebras’, Czechoslovak Math. J. 52 (2002), 255273.

[25]A. Ramsay , ‘Dimension theory in complete weakly modular orthocomplemented lattices’, Trans. Amer. Math. Soc. 116 (1965), 931.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords: