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On the de Morgan Property of the Standard Brouwer–Zadeh Poset

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Abstract

The standard Brouwer–Zadeh poset Σ(H) is the poset of all effect operators on a Hilbert space H, naturally equipped with two types of orthocomplementation. In developing the theory, the question occured if (when) Σ(H) fulfils the de Morgan property with respect to both orthocomplementation operations. In Ref.3 the authors proved that it is the case provided dimH<∞, and they conjectured that if dimH=∞, then the answer is in the negative. In this note, we first give a somewhat simpler proof of the known result for dimH<∞, and then we give a proof to the conjecture: We show that if dimH=∞, then the de Morgan property is not valid.

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Authors and Affiliations

  1. Dipartimento di Informatica, Sistemistica e Comunicazione, Università di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, I-20126, Milano, Italy;

    G. Cattaneo

  2. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Zikova 4, 16627, Praha 6, Czech Republic;

    J. Hamhalter

  3. Department of Mathematics, Faculty of Electrical Engineering, Czech Technical University, Zikova 4, 16627, Praha 6, Czech Republic;

    P. Pták

Authors
  1. G. Cattaneo
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  2. J. Hamhalter
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  3. P. Pták
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Cattaneo, G., Hamhalter, J. & Pták, P. On the de Morgan Property of the Standard Brouwer–Zadeh Poset. Foundations of Physics 30, 1801–1805 (2000). https://doi.org/10.1023/A:1026414704133

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  • DOI: https://doi.org/10.1023/A:1026414704133

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