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Intuitionistic Fuzzy Sets Generated by Archimedean Metrics and Ultrametrics

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

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.

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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

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