International Journal of Theoretical Physics

, Volume 52, Issue 6, pp 2171–2180 | Cite as

Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions



In (Riečanová and Zajac in Rep. Math. Phys. 70(2):283–290, 2012) it was shown that an effect algebra E with an ordering set \(\mathcal{M}\) of states can by embedded into a Hilbert space effect algebra \(\mathcal{E}(l_{2}(\mathcal{M}))\). We consider the problem when its effect algebraic MacNeille completion \(\hat{E}\) can be also embedded into the same Hilbert space effect algebra \(\mathcal {E}(l_{2}(\mathcal{M}))\). That is when the ordering set \(\mathcal{M}\) of states on E can be extended to an ordering set of states on \(\hat{E}\). We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.


Effect algebra MV-effect algebra MacNeille completion Positive linear operators in Hilbert space Hilbert space effect-representation 



Jiří Janda kindly acknowledges the support by Masaryk University, grant 0964/2009 and ESF Project CZ.1.07 /2.3.00/20.0051 Algebraic Methods in Quantum Logic of the Masaryk University.

Zdenka Riečanová kindly acknowledges the support by the Science and Technology Assistance Agency under the contract APVV-0178-11 Bratislava SR, and VEGA-grant of MŠ SR No. 1/0297/11.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics, Faculty of ScienceMasaryk UniversityBrnoCzech Republic
  2. 2.Department of Mathematics, Faculty of Electrical Engineering and Information TechnologySlovak University of TechnologyBratislavaSlovak Republic

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