University of Sistan and Baluchestan Iranian Journal of Fuzzy Systems 1735-0654 23 4 2026 08 01 Characterizing idempotent uninorms on a bounded chain 69 74 10011 10.22111/ijfs.2026.10011 EN Y. Su School of Mathematics Science, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, China W. Zong School of Mathematical Sciences, University of Jinan, Jinan, 250022, China R. Mesiar Palack´y University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, Olomouc, 771 46, Czech Republic Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´eho 11, 810 05 Bratislava Slovakia Journal Article 2026 07 08 Completeness is essential for characterizing idempotent uninorms on complete chains, as it guarantees the welldefinedness of their corresponding characterization functions. In the case of a general chain, one cannot define the<br />aforementioned characterization functions by taking the supremum or infimum of a prescribed subset. When constructing<br />the real numbers via the Dedekind completion of the rationals, each rational number is associated with a<br />rational cut, which forms a down-set. Inspired by this line of reasoning, this paper provides a direct characterization<br />of idempotent uninorms defined on bounded chains via decreasing symmetric set-valued functions that map the chain<br />to its family of down-sets. https://ijfs.usb.ac.ir/article_10011_3ab4ff6d4e857bf3e3f3dd258031576e.pdf