Characterizing idempotent uninorms on a bounded chain

Document Type : Research Paper

Authors

1 School of Mathematics Science, Suzhou University of Science and Technology, Suzhou, Jiangsu 215009, China

2 School of Mathematical Sciences, University of Jinan, Jinan, 250022, China

3 Palack´y University Olomouc, Faculty of Science, Department of Algebra and Geometry, 17. listopadu 12, Olomouc, 771 46, Czech Republic

4 Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´eho 11, 810 05 Bratislava Slovakia

Abstract

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
aforementioned characterization functions by taking the supremum or infimum of a prescribed subset. When constructing
the real numbers via the Dedekind completion of the rationals, each rational number is associated with a
rational cut, which forms a down-set. Inspired by this line of reasoning, this paper provides a direct characterization
of idempotent uninorms defined on bounded chains via decreasing symmetric set-valued functions that map the chain
to its family of down-sets.

Keywords

References

[1] M. Couceiro, J. Devillet, J. L. Marichal, Characterizations of idempotent discrete uninorms, Fuzzy Sets and Systems,
334 (2018), 60-72. https://doi.org/10.1016/j.fss.2017.06.013
[2] E. Czoga la, J. Drewniak, Associative monotonic operations in fuzzy set theory, Fuzzy Sets and Systems, 12(3)
(1984), 249-269. https://doi.org/10.1016/0165-0114(84)90072-1
[3] B. A. Davey, H. A. Priestley, Introduction to lattices and order, Cambridge University Press, 2012. https://doi.
org/10.1017/CBO9780511809088
[4] B. De Baets, Idempotent uninorms, European Journal of Operational Research, 118(3) (1999), 631-642. https:
//doi.org/10.1016/S0377-2217(98)00325-7
[5] B. De Baets, J. C. Fodor, D. Ruiz-Aguilera, J. Torrens, Idempotent uninorms on finite ordinal scales, International
Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 17(1) (2009), 1-14. https://doi.org/10.1142/
S021848850900570X
[6] J. Golan, The theory of semirings with applications in mathematics and theoretical computer science, Pitman Monographs and Surveys in Pure and Applied Mathematics, 54, Longman Scientific and Technical, 1992.
[7] M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap, Aggregation functions, Cambridge University Press, 2009. https:
//doi.org/10.1017/CBO9781139644150
[8] F. Kara¸cal, R. Mesiar, Uninorms on bounded lattices, Fuzzy Sets and Systems, 261 (2015), 33-43. https://doi.
org/10.1016/j.fss.2014.05.001
[9] J. Mart´ın, G. Mayor, J. Torrens, On locally internal monotonic operations, Fuzzy Sets and Systems, 137(1) (2003),
27-42. https://doi.org/10.1016/S0165-0114(02)00430-X
[10] M. Mas, S. Massanet, D. Ruiz-Aguilera, J. Torrens, A survey on the existing classes of uninorms, Journal of
Intelligent Fuzzy Systems, 29(3) (2015), 1021-1037. https://doi.org/10.3233/IFS-151728
[11] R. Mesiar, A. Koles´arov´a, A. Stupˇnanov´a, Quo vadis aggregation?, International Journal of General Systems, 47(2)
(2018), 97-117. https://doi.org/10.1080/03081079.2017.1402893
[12] A. Mesiarov´a-Zem´ankov´a, Characterization of uninorms with continuous underlying t-norm and t-conorm by their
set of discontinuity points, IEEE Transactions on Fuzzy Systems, 26(2) (2018), 705-714. https://doi.org/10.
1109/TFUZZ.2017.2688346
[13] Y. Ouyang, H. Zhang, Z. Wang, B. De Baets, Idempotent uninorms on a complete chain, Fuzzy Sets and Systems,
448 (2022), 107-126. https://doi.org/10.1016/j.fss.2022.03.003
[14] W. Rudin, Principles of mathematical analysis, (3rd ed.), McGraw-Hill Book Co. Inc, 1976.
[15] D. Ruiz-Aguilera, J. Torrens, B. De Baets, J. Fodor, Some remarks on the characterization of idempotent uninorms,
in E. H¨ullermeier, R. Kruse, and F. Hoffmann (Eds.), Computational Intelligence for Knowledge-Based Systems
Design. IPMU 2010. Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, 6178 (2010), 425-434. https:
//doi.org/10.1007/978-3-642-14049-5_44
[16] Y. Su, A. Mesiarov´a-Zem´ankov´a, R. Mesiar, Idempotent uninorms on a bounded chain, Fuzzy Sets and Systems,
471 (2023), 108671. https://doi.org/10.1016/j.fss.2023.108671
[17] Y. Su, W. Zong, A. Mesiarov´a-Zem´ankov´a, R. Mesiar, B. De Baets, A state-of-the-art survey of the most prominent
classes of uninorms on the unit interval, Fuzzy Sets and Systems, 519 (2025), 109518. https://doi.org/10.1016/
j.fss.2025.109518
[18] R. R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80(1) (1996), 111-120. https:
//doi.org/10.1016/0165-0114(95)00133-6
Iranian Journal of Fuzzy Systems
Volume 23, Issue 4
July and August 2026
Pages 69-74

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