Intuitionistic Fuzzy Sets Generated by Archimedean Metrics and Ultrametrics

  • Peter VassilevEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 657)


For a nonempty universe E it is shown that the standard intutitionistic fuzzy sets (IFSs) over E are generated by Manhattan metric. For several other types of intuitionistic fuzzy sets the metrics, generating them, are found. As a result a general metric approach is developed. For a given abstract metric d,  the corresponding objects are called d-intuitionistic fuzzy sets. Special attention is given to the case when d is a metric generated by a subnorm. If d is generated by an absolute normalized norm (the Archimedean case), an important result is established: the class of all d-intuitionistic fuzzy sets over E is isomorphic (in the sense of bijection) to the class of all IFSs over E. In § 4, instead of \(\mathbb {R}^2,\) the Cartesian product \(\mathbb {Q}^2,\) of the rational number field \(\mathbb {Q}\) with itself, is considered. It is shown that \(\mathbb {Q}^2\) may be transformed in infinitely many ways (depending on family of primes p) into a field with non-Archimedean field norm \(\varPhi _p\) generated by p-adic norm. Using the corresponding ultrametric \(d_{\varPhi _p}\) on \(\mathbb {Q}^2,\) objects called \(d_{\varPhi _p}\)-intuitionistic fuzzy sets over E are defined (the non-Archimedean case). Thus, for the first time intuitionistic fuzzy sets depending on ultrametric are introduced.


  1. 1.
    Atanassov, K.: Intuitionistic Fuzzy Sets. VII ITKR’s session (deposed in Central Sci. -Techn. Library of Bulg. Acad. of Sci. 1697/84) Sofia (1983) (in Bulgarian)Google Scholar
  2. 2.
    Atanassov, K.: A second type of intuitionistic fuzzy sets. BUSEFAL 56, 66–70 (1993)Google Scholar
  3. 3.
    Atanassov, K.: Intuitionistic Fuzzy Sets. Springer Physica-Verlag, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  4. 4.
    Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer Physica-Verlag, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bingham, N.H., Ostaszewski, A.J.: Normed groups: dichotomy and duality. LSE-CDAM Report, LSE-CDAM-2008-10revGoogle Scholar
  6. 6.
    Bonsall, F., Duncan, J.: Numerical Ranges II. London Mathematical Society. Lecture Notes Series, vol. 10 (1973)Google Scholar
  7. 7.
    Bullen, P.S.: Handbook of Means and their Inequalities. Kluwer Academic Publishers, Dordrecht (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Deza, M., Deza, E.: Encyclopedia of Distances. Springer, Heidelberg (2009)CrossRefzbMATHGoogle Scholar
  9. 9.
    Dougherty, G.: Pattern Recognition and Classification an Introduction. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Ireland, K., Rosen, M.: Classical Introduction to Modern Number Theory. Springer Physica-Verlag, New York (1990)CrossRefzbMATHGoogle Scholar
  11. 11.
    Koblitz, N.: \(P\)-adic Numbers, \(p\)-adic Analysis, and Zeta-Functions, 2nd edn. Springer, New York (1984)CrossRefzbMATHGoogle Scholar
  12. 12.
    Körner, M.-C.: Minisum Hyperspheres. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  13. 13.
    Krause, E.F.: Taxicab Geometry. Dover Publications, New York (1975)Google Scholar
  14. 14.
    Palaniapan, N., Srinivasan, R., Parvathi, R.: Some operations on intuitionistic fuzzy sets of root type. NIFS 12(3), 20–29 (2006)Google Scholar
  15. 15.
    Parvathi, R., Vassilev, P., Atanassov, K.: A note on the bijective correspondence between intuitionistic fuzzy sets and intuitionistic fuzzy sets of \(p\)-th type. In: New Developments in Fuzzy Sets, Intuitionistic Fuzzy Sets, Generalized Nets and Related Topics. Volume I: Foundations, SRI PAS IBS PAN, Warsaw, pp. 143–147 (2012)Google Scholar
  16. 16.
    Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. I. Springer, Berlin (1976)Google Scholar
  17. 17.
    Vassilev, P., Parvathi, R., Atanassov, K.: Note On Intuitionistic Fuzzy Sets of \(p\)-th Type. Issues in Intuitionistic Fuzzy Sets and Generalized Nets 6, 43–50 (2008)Google Scholar
  18. 18.
    Vassilev, P.: A Metric Approach To Fuzzy Sets and Intuitionistic Fuzzy Sets. In: Proceedings of First International Workshop on IFSs, GNs, KE, pp. 31–38 (2006)Google Scholar
  19. 19.
    Vassilev, P.: Operators similar to operators defined over intuitionistic fuzzy sets. In: Proceedings of 16th International Conference on IFSs, Sofia, 910 Sept. 2012. Notes on Intuitionistic Fuzzy Sets, vol. 18, No. 4, 40–47 (2012)Google Scholar
  20. 20.
    Vassilev-Missana, M., Vassilev, P.: On a Way for Introducing Metrics in Cartesian Product of Metric Spaces. Notes on Number Theory and Discrete Mathematics 8(4), 125–128 (2002)Google Scholar
  21. 21.
    Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Bioinformatics and Mathematical Modelling DepartmentInstitute of Biophysics and Biomedical Engineering, Bulgarian Academy of SciencesSofiaBulgaria

Personalised recommendations