Fuzzy order theory: characterization and implementation

Abstract

This paper develops a new fuzzy extension of the classical Knaster–Tarski fixed point theorem and the converse result of Anne C. Davis. Working within the framework of r-fuzzy ordered sets, we introduce the notion of r-fuzzy complete lattices and establish necessary and sufficient conditions for the existence of fixed points of r-fuzzy monotone mappings. We prove that every r-fuzzy monotone self-map on a non-empty r-fuzzy complete lattice admits both a greatest and a least fixed point. Furthermore, we obtain a fuzzy version of Davis’s characterization of complete lattices by constructing an explicit r-fuzzy monotone operator that fails to have fixed points when completeness is absent. These results provide a unified approach to fixed-point theory in fuzzy environments and extend several known theorems in both classical and fuzzy order theory.