Abstract
In mathematics, distance and similarity are known as dual concepts. However, the concept of similarity is interpreted as fuzzy similarity or T-equivalence relation, where T is a triangular norm (t-norm in brief), when we discuss a fuzzy environment. Dealing with multi-polarity in practical examples with fuzzy data leadsus to introduce a new concept called m-polar T-equivalence relations based on a finitely multivalued t-norm T, and to study the metric behavior of such relations. First, we study the new operators including the m-polar triangular norm T and conorm S as well as m-polar implication I and m-polar negation N, acting on the Cartesian product of [0,1]m-times.Then, using the m-polar negations N, we provide a method to construct a new type of metric spaces, called m-polar S-pseudo-ultrametric, from the m-polar T-equivalences, and reciprocally for constructing m-polar T-equivalences based on the m-polar S-pseudo-ultrametrics. Finally, the link between fuzzy graphs and m-polar S-pseudo-ultrametrics is considered. An algorithm is designed to plot a fuzzy graph based on the m-polar SL-pseudo-ultrametric, where SL is the m-polar Lukasiewicz t-conorm, and is illustrated by a numerical example which verifies our method.
Highlights
The pairwise comparison and classification of objects is one of the main steps in any field dealing with data analysis. This task is generally handled by equivalence relations for crisp data and T-equivalences in fuzzy environments, where T is a triangular norm called briefly a t-norm [1,2]
In fuzzy environments, the T-equivalences usually produce pseudo-ultrametrics, rather than standard metric spaces, in which the triangular inequality of a metric has been generalized by the maximum operator [3,4,5,6]
We aim to explore in greater depth the one-to-one correspondence between m-polar T-equivalences and m-polar S-pseudo-(ultra)metrics in Section 4 by using m-polar negations
Summary
The pairwise comparison and classification of objects is one of the main steps in any field dealing with data analysis. M-times, an extension of the classical fuzzy logic into m-polar fuzzy logic is required In such a new multivalued logic, we need to study the concepts of conjunction (usually interpreted by t-norms), disjunction (defined by t-conorms), implication, and negation for the m-polar case. Because the maximum operator is a t-conorm, it seems natural to consider a stronger version of pseudo-ultrametric for the m-polar case as a S-pseudo-ultrametric where S is any m-polar t-conorm It can be determined whether or not the result is always an m-polar pseudo-(ultra)metric if the traditional concept of fuzzy duality, which is used to define the induced metrics by T-equivalences, is replaced by an m-polar negation N. An algorithm is designed to compute a fuzzy graph based on the m-polar S L -pseudo-ultrametric, where S L is the m-polar Lukasiewicz t-conorm, and we illustrate it with an example
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