Regular orders on triangular fuzzy numbers and the weak law of trichotomy

Abstract: Building upon specific compatibility conditions, we establish fundamental structural results concerning ordering relations for triangular fuzzy numbers. We demonstrate that orders satisfying compatibility with arithmetic operations, MIN-MAX operators, and the Weak Law of Trichotomy (WLT) are completely determined on the fibers of the natural projection to real numbers. Furthermore, such orders naturally induce - in analogy with real numbers - well-defined notions of fuzzy absolute value and fuzzy distance that preserve the essential properties of their classical counterparts. These results enable us to characterize open and closed balls through interval representations, providing a robust theoretical framework for future studies regarding metric properties of fuzzy numbers.