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Varieties of Orthocomplemented Lattices Induced by Łukasiewicz-Groupoid-Valued Mappings

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Abstract

In the logico-algebraic approach to the foundation of quantum mechanics we sometimes identify the set of events of the quantum experiment with an orthomodular lattice (“quantum logic”). The states are then usually associated with (normalized) finitely additive measures (“states”). The conditions imposed on states then define classes of orthomodular lattices that are sometimes found to be universal-algebraic varieties. In this paper we adopt a conceptually different approach, we relax orthomodular to orthocomplemented and we replace the states with certain subadditive mappings that range in the Łukasiewicz groupoid. We then show that when we require a type of “fulness” of these mappings, we obtain varieties of orthocomplemented lattices. Some of these varieties contain the projection lattice in a Hilbert space so there is a link to quantum logic theories. Besides, on the purely algebraic side, we present a characterization of orthomodular lattices among the orthocomplemented ones. - The intention of our approach is twofold. First, we recover some of the Mayet varieties in a principally different way (indeed, we also obtain many other new varieties). Second, by introducing an interplay of the lattice, measure-theoretic and fuzzy-set notions we intend to add to the concepts of quantum axiomatics.

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Authors and Affiliations

  1. Ruská 22 101 00 Prague 10, Prague, Czech Republic

    Milan Matoušek

  2. Department of Mathematics, Czech Technical University, Faculty of Electrical Engineering, 166 27 Prague 6, Prague, Czech Republic

    Pavel Pták

Authors
  1. Milan Matoušek
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  2. Pavel Pták
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Correspondence to Milan Matoušek.

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Matoušek, M., Pták, P. Varieties of Orthocomplemented Lattices Induced by Łukasiewicz-Groupoid-Valued Mappings. Int J Theor Phys 56, 4004–4016 (2017). https://doi.org/10.1007/s10773-017-3411-x

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  • DOI: https://doi.org/10.1007/s10773-017-3411-x

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