Abstract
In this note, we present an alternative approach to the concept of a fuzzy metric, calling it a revised fuzzy metric. In contrast to the traditional approach to the theory of fuzzy metric spaces which is based on the use of a t-norm, we proceed from a t-conorm in the definition of a revised fuzzy metric. Here, we restrict our study to the case of fuzzy metrics as they are defined by George-Veeramani, however, similar revision can be done also for some other approaches to the concept of a fuzzy metric.
Highlights
Introduction and MotivationIn 1951, Menger introduced the concept of a statistical metric [1]
We have defined an RGV-fuzzy metric as the dual version of a GV-fuzzy metric and illustrated how basic concepts and results of the theory of GV-fuzzy metric spaces can be reformulated and reproved in the terms of RGV-fuzzy metric spaces
It was not our aim to convince the reader that RGV-fuzzy metrics are “better” than GV-fuzzy metrics
Summary
Based on the concept of a statistical metric, Kramosil and Michalek introduced the notion of a fuzzy metric in [2]. In 1994, George and Veeramani [3], see [4], slightly modified the original concept of a KM-fuzzy metric—we call this modification by a GV-fuzzy metric This modification allows many natural examples of fuzzy metrics, in particular, fuzzy metrics constructed from metrics. The fuzzy distance between two equal points is 1, while in cases when one point is “far” from the other, the fuzzy distance between them is “close” to 0 This may look strange if only one does not think of a fuzzy metric as the counterpart of a statistical metric. A similar revision can be done for KM-fuzzy metrics as well
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