Advertisement

Studia Logica

, Volume 82, Issue 2, pp 245–270 | Cite as

MV-Algebras and Quantum Computation

  • Antonio Ledda
  • Martinvaldo Konig
  • Francesco Paoli
  • Roberto Giuntini
Article

Abstract

We introduce a generalization of MV algebras motivated by the investigations into the structure of quantum logical gates. After laying down the foundations of the structure theory for such quasi-MV algebras, we show that every quasi-MV algebra is embeddable into the direct product of an MV algebra and a “flat” quasi-MV algebra, and prove a completeness result w.r.t. a standard quasi-MV algebra over the complex numbers.

Keywords

fuzzy logic MV algebra quantum computation quantum logic quasi-MV algebra residuated structures 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aglianò, P., and A. Ursini, ‘On subtractive varieties II: General properties’, Algebra Universalis, 36 (1996), 222–259.Google Scholar
  2. [2]
    Cattaneo, G., M. L. Dalla Chiara, R. Giuntini, and R. Leporini, ‘An unsharp logic from quantum computation’, International Journal of Theoretical Physics, 43, 7–8 (2004), 1803–1817.Google Scholar
  3. [3]
    Cattaneo, G., M. L. Dalla Chiara, R. Giuntini, and R. Leporini, ‘Quantum computational structures’, Mathematica Slovaca, 54 (2004), 87–108.Google Scholar
  4. [4]
    Chang, C. C., ‘A new proof of the completeness of Lukasiewicz axioms’, Transactions of the American Mathematical Society, 93 (1959), 74–90.Google Scholar
  5. [5]
    Cignoli, R., I. M. L. D’Ottaviano, and D. Mundici, Algebraic Foundations of Many-Valued Reasoning, Kluwer, Dordrecht, 1999.Google Scholar
  6. [6]
    Dalla Chiara, M. L., R. Giuntini, and R. Greechie, Reasoning in Quantum Theory, Kluwer, Dordrecht, 2004.Google Scholar
  7. [7]
    Dalla Chiara, M. L., R. Giuntini, and R. Leporini, ‘Quantum computational logics: A survey’, in V. F. Hendricks, J. Malinowski (eds.), Trends in Logic: 50 Years of Studia Logica, Kluwer, Dordrecht, 2003, pp. 213–255.Google Scholar
  8. [8]
    Gumm, H. P., and A. Ursini, ‘Ideals in universal algebra’, Algebra Universalis, 19 (1984), 45–54.Google Scholar
  9. [9]
    Hagermann, J., and C. Hermann, ‘A concrete ideal multiplication for algebraic systems and its relation to congruence distributivity’, Archive for Mathematics, 32 (1979), 234–245.Google Scholar
  10. [10]
    Idziak, P. M., ‘Lattice operations in BCK algebras’, Mathematica Japonica, 29, 6 (1984), 839–846.Google Scholar
  11. [11]
    Maltsev, A. I., ‘On the general theory of algebraic systems’(in Russian), Mat. Sb. (N. S.), 35, 77 (1954), 3–20.Google Scholar
  12. [12]
    Mitschke, A., ‘Implication algebras are 3-permutable and 3-distributive’, Algebra Universalis, 1 (1971), 182–186.Google Scholar
  13. [13]
    Paoli, F., Substructural Logics: A Primer, Kluwer, Dordrecht, 2002.Google Scholar
  14. [14]
    Ursini, A., ‘On subtractive varieties I’, Algebra Universalis, 31 (1994), 204–222.CrossRefGoogle Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Antonio Ledda
    • 1
  • Martinvaldo Konig
    • 1
  • Francesco Paoli
    • 1
  • Roberto Giuntini
    • 1
  1. 1.Department of EducationUniversity of CagliariCagliariItaly

Personalised recommendations