Sequential ordering relations with application to fuzzy numbers

Abstract

The ranking of fuzzy numbers has become a challenging task in fuzzy set theory due to their complex, multi-dimensional nature. While the Klir-Yuan partial order provides a natural term-wise comparison of $α$-cuts, it often leaves many fuzzy numbers incomparable. To address this, various ranking methods have been developed to construct total preorders between them. However, many classical approaches suffer from significant information loss as they imply a defuzzification process. On the other hand, approaches such as admissible orders allow defining total orders, but at the expense of imposing strict algebraic rules that may contradict human intuition. In this study, we introduce a generalized sequential ordering framework to overcome these limitations. By establishing a sequence space over a totally preordered base space, we construct a flexible lexicographical structure that sequentially resolves ties. We prove that this framework yields total preorders and, under injectivity conditions, total orders. Furthermore, we analyze the compatibility of these sequential orders with the notion of admissibility. We also show that our proposed framework provides a unified mathematical umbrella that encompasses and generalizes existing ranking techniques, offering highly discriminative ordering relations for fuzzy numbers and beyond.