Intuitionistic Fuzzy Sets Generated by Archimedean Metrics and Ultrametrics

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.