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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.

Revision of Pseudo-Ultrametric Spaces Based on m-Polar T-Equivalences and Its Application in Decision Making