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.
Copyright © 2026 Amine Faiz, Khadija Bouzkoura, Adil Baiz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Cite this article
A. Faiz, K. Bouzkoura, A. Baiz, Fuzzy order theory: characterization and implementation, Adv. Fixed Point Theory 16 (2026), 11. https://doi.org/10.28919/afpt/9660
BibTeX
@article{Faiz2026,
author = {Faiz, Amine and Bouzkoura, Khadija and Baiz, Adil},
title = {Fuzzy order theory: characterization and implementation},
journal = {Adv. Fixed Point Theory},
volume = {16},
year = {2026},
pages = {11},
doi = {10.28919/afpt/9660}
}