Regular orders on triangular fuzzy numbers and the weak law of trichotomy
Jaime Cesar dos Santos Filho
TL;DR
The paper addresses how to order triangular fuzzy numbers under structurally desirable conditions by introducing regular orders compatible with arithmetic, MIN-MAX, and the weak law of trichotomy (WLT). It proves these orders are determined fiberwise via the natural projection to real numbers and gives rise to well-behaved fuzzy absolute value and distance, enabling interval descriptions of balls. When adding projection and MIN-MAX compatibility, only two global regular orders remain: the upper-sum and lower-sum orders, with the upper-sum uniquely supporting a fuzzy absolute value. Theoretical results are complemented by numerical comparisons of existing ranking methods, illustrating the practical implications for fuzzy metric properties and decision-making contexts.
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.