mathematics Article Revision of Pseudo-Ultrametric Spaces Based on m-Polar T-Equivalences and Its Application in Decision Making Azadeh Zahedi Khameneh 1, * , Adem Kilicman 1,2 and Fadzilah Md Ali 1,2   Citation: Zahedi Khameneh, A.; Kilicman, A.; Md Ali, F. Revision of Pseudo-Ultrametric Spaces Based on m-Polar T-Equivalences and Its Application in Decision Making. Mathematics 2021, 9, 1232. https:// doi.org/10.3390/math9111232 Academic Editor: Daniel Gómez Gonzalez Received: 10 March 2021 Accepted: 13 May 2021 Published: 28 May 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). 1 Institute for Mathematical Research, Universiti Putra Malaysia, Serdang, Selangor 43400, Malaysia; akilic@upm.edu.my (A.K.); fadzilahma@upm.edu.my (F.M.A.) 2 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, Serdang, Selangor 43400, Malaysia * Correspondence: zk.azadeh@upm.edu.my or azadeh503@gmail.com 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 S L -pseudo-ultrametric, where S L is the m-polar Lukasiewicz t-conorm, and is illustrated by a numerical example which verifies our method. Keywords: m-polar fuzzy relation; m-polar T-equivalence; m-polar S-pseudo-ultrametric; fuzzy graph; group decision making 1. Introduction 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]. Traditionally, the equivalence relations (known also as in- distinguishability relations) and (pseudo-)metrics (or distinguishability relations) have a close link, where one can be defined as the dual of the other. However, in fuzzy envi- ronments, 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 [3–6]. Recently, the notion of multi-polarity, arising from multi-source data, has been in- troduced in the fuzzy context to achieve higher accuracy in data analysis. However, this concept has two sides. From the first perspective, it describes fuzzy rule-based classification systems, designed based on different categories, where the final rule (which consists of a set of fuzzy rules) contains the consequent class of the final rule from the m predefined classes/categories/patterns [7–10]. From another direction, multi-polarity provides a flex- ible framework for input and output data in information systems, applied in computer science, multi-criteria decision making, and graph theory, where instead of one absolute value u ∈ [0, 1] the m-tuple u =(u 1 , ··· , u m ) ∈ [0, 1] ×···× [0, 1] is employed to describe data based on different parameters/criteria [11–13]. Note that this paper follows the second case, which is known as m-polar fuzzy set theory. Mathematics 2021, 9, 1232. https://doi.org/10.3390/math9111232 https://www.mdpi.com/journal/mathematics