Abstract
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Here and further iff means “if and only if”.
- 2.
- 3.
Manhattan norm must be called Hamming norm only when the components of x are binary.
- 4.
A more precise denotation would be \({{}^{(\mathbb {R}^2)}}d\)-IFS but we will omit it since there is no danger of misunderstanding.
- 5.
This definition remains valid if \(\mathbb {Q}\) is replaced by arbitrary field.
- 6.
We remind that \(\varPhi \) is a normalized norm, if \(\varPhi ((1,0))=\varPhi ((0,1))=1\).
References
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)
Atanassov, K.: A second type of intuitionistic fuzzy sets. BUSEFAL 56, 66–70 (1993)
Atanassov, K.: Intuitionistic Fuzzy Sets. Springer Physica-Verlag, Heidelberg (1999)
Atanassov, K.: On Intuitionistic Fuzzy Sets Theory. Springer Physica-Verlag, Heidelberg (2012)
Bingham, N.H., Ostaszewski, A.J.: Normed groups: dichotomy and duality. LSE-CDAM Report, LSE-CDAM-2008-10rev
Bonsall, F., Duncan, J.: Numerical Ranges II. London Mathematical Society. Lecture Notes Series, vol. 10 (1973)
Bullen, P.S.: Handbook of Means and their Inequalities. Kluwer Academic Publishers, Dordrecht (2003)
Deza, M., Deza, E.: Encyclopedia of Distances. Springer, Heidelberg (2009)
Dougherty, G.: Pattern Recognition and Classification an Introduction. Springer, New York (2013)
Ireland, K., Rosen, M.: Classical Introduction to Modern Number Theory. Springer Physica-Verlag, New York (1990)
Koblitz, N.: \(P\)-adic Numbers, \(p\)-adic Analysis, and Zeta-Functions, 2nd edn. Springer, New York (1984)
Körner, M.-C.: Minisum Hyperspheres. Springer, Heidelberg (2011)
Krause, E.F.: Taxicab Geometry. Dover Publications, New York (1975)
Palaniapan, N., Srinivasan, R., Parvathi, R.: Some operations on intuitionistic fuzzy sets of root type. NIFS 12(3), 20–29 (2006)
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)
Pólya, G., Szegö, G.: Problems and Theorems in Analysis, vol. I. Springer, Berlin (1976)
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)
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)
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)
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)
Zadeh, L.: Fuzzy sets. Inf. Control 8(3), 338–353 (1965)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Vassilev, P. (2017). Intuitionistic Fuzzy Sets Generated by Archimedean Metrics and Ultrametrics. In: Sgurev, V., Yager, R., Kacprzyk, J., Atanassov, K. (eds) Recent Contributions in Intelligent Systems. Studies in Computational Intelligence, vol 657. Springer, Cham. https://doi.org/10.1007/978-3-319-41438-6_19
Download citation
DOI: https://doi.org/10.1007/978-3-319-41438-6_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41437-9
Online ISBN: 978-3-319-41438-6
eBook Packages: EngineeringEngineering (R0)