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

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Jiří Janda

  2. Department of Mathematics, Faculty of Electrical Engineering and Information Technology, Slovak University of Technology, Ilkovičova 3, 812 19, Bratislava, Slovak Republic

    Zdenka Riečanová

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  1. Jiří Janda
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  2. Zdenka Riečanová
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Correspondence to Jiří Janda.

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Janda, J., Riečanová, Z. Extensions of Ordering Sets of States from Effect Algebras onto Their MacNeille Completions. Int J Theor Phys 52, 2171–2180 (2013). https://doi.org/10.1007/s10773-013-1532-4

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  • DOI: https://doi.org/10.1007/s10773-013-1532-4

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