Fuzzy Metrics in Terms of Fuzzy Relations

In this paper, we study the concept of fuzzy metrics from the perspective of fuzzy relations. Specifically, we analyze the commonly used definitions of fuzzy metrics. We begin by noting that crisp metrics can be uniquely characterized by linear order relations. Further, we explore the criteria that crisp relations must satisfy in order to determine a crisp metric. Subsequently, we extend these conditions to obtain a fuzzy metric and investigate the additional axioms involved. Additionally, we introduce the definition of an extensional fuzzy metric or E-d-metric, which is a fuzzification of the expression (Formula presented.). Thus, we examine fuzzy metrics from both the linear order and from the equivalence relation perspectives, where one argument is a value (Formula presented.) and the other is a number within the range (Formula presented.).