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States with values in the Łukasiewicz groupoid

  • Milan Matoušek EMAIL logo and Pavel Pták EMAIL logo
Published/Copyright: July 4, 2016
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 66 Issue 2

Abstract

In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let ∗ stand for the Łukasiewicz t-norm on [0, 1]2. Let us consider the operation ⋄ on [0, 1] by setting x ⋄ y = (x⊥ ∗y⊥)⊥ ∗ (x∗y)⊥, where x⊥ = 1−x. Let us call the triple L = ([0, 1], ⋄, 1) the Łukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z2-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as “quantum logics” in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state.

Keywords: Łukasiewicz t-norm; Łukasiewicz groupoid; Boolean algebra; orthomodular poset; quantum logic; classical and Z2-valued state; Greechie paste job
MSC 2010: Primary 03E72; 28B99; 81P10

To Anatolij Dvurečenskij with best wishes

(Communicated by Sylvia Pulmannová)

References

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Received: 2014-2-12
Accepted: 2014-3-17
Published Online: 2016-7-4
Published in Print: 2016-4-1

© 2016 Mathematical Institute Slovak Academy of Sciences

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https://doi.org/10.1515/ms-2015-0139
Keywords for this article
Łukasiewicz t-norm; Łukasiewicz groupoid; Boolean algebra; orthomodular poset; quantum logic; classical and Z2-valued state; Greechie paste job

Articles in the same Issue

  1. Research Article
  2. Holistic logical arguments in quantum computation
  3. Research Article
  4. States with values in the Łukasiewicz groupoid
  5. Research Article
  6. On realization of effect algebras
  7. Research Article
  8. States on symmetric logics: extensions
  9. Research Article
  10. Extensions and measurability in quantum measure spaces
  11. Research Article
  12. Convex subalgebras of upper bounded GMV-algebras
  13. Research Article
  14. Combining boolean algebras and ℓ-groups in the variety generated by chang’s mv-algebra
  15. Research Article
  16. Large rigid sets of algebras with respect to embeddability
  17. Research Article
  18. Conformal algebras, vertex algebras, and the logic of locality
  19. Research Article
  20. Extension of measures on pseudo-D-lattices
  21. Research Article
  22. Note on a parameter switching method for nonlinear ODEs
  23. Research Article
  24. Compatibility for probabilistic theories
  25. Research Article
  26. Completeness of Gelfand-Neumark-Segal inner product space on Jordan algebras
  27. Research Article
  28. Commutativity in a synaptic algebra
  29. Research Article
  30. Discrete averaged mixing applied to the logarithmic distributions
  31. Research Article
  32. Automorphisms of decompositions
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