Ordinal sum constructions for aggregation functions on the real unit interval

Document Type : Research Paper

Authors

1 Mathematical Institute, Slovak Academy of Sciences, 814 73 Bratislava, Slovakia

2 Faculty of Civil Engineering, Department of Mathematics, Slovak University of Technology, 81368 Bratislava, Slovakia

3 School of Mathematics Sciences, Suzhou University of Science and Technology, Suzhou 215009, China

Abstract

We discuss ordinal sums as one of powerful tools in the aggregation theory serving, depending on the context, both as a construction method and as a representation, respectively. Up to recalling of several classical results dealing with ordinal sums, in particular dealing, e.g.,  with continuous t-norms, copulas, or recent results, e.g., concerning uninorms with continuous underlying functions, we present also several new results, such as the uniqueness of the link between t-norms or t-conorms, and related Archimedean components, problems dealing with the cardinality of the considered index sets in ordinal sums, or infinite ordinal sums of aggregation functions covering by one type of ordinal sums both t-norms and t-conorms ordinal sums.

Keywords

References

[1] P. Akella, Structure of n-uninorms, Fuzzy Sets and Systems, 158(15) (2007), 1631-1651.
[2] C. Alsina, M. J. Frank, B. Schweizer, Associative functions: Triangular norms and copulas, World Scientific, Singapore, 2006.
[3] M. Baczyński, P. Drygaś, A. Król, R. Mesiar, New types of ordinal sum of fuzzy implications, 2017 IEEE International Conference on Fuzzy Systems, (2017), 1-6.
[4] M. Baczyński, P. Drygaś, R. Mesiar, Monotonicity in the construction of ordinal sums of fuzzy implications, In Proc. of AGOP 2017, (2017), 189-199.
[5] B. De Baets, R. Mesiar, Residual implicators of continuous t-norms, In Proc. of EUFIT ’96, Zimmermann H. J. (editor), ELITE, (Aachen 1996), (1996), 27-31.
[6] B. De Baets, R. Mesiar, Ordinal sums of aggregation operators, In: Bouchon-Meunier B et al. (eds), Technologies for Constructing Intelligent Systems 2, Physica-Verlag, Heidelberg, (2002), 137-148.
[7] G. Beliakov, A. Pradera, T. Calvo, Aggregation functions: A guide for practitioners, New York, Springer-Verlag, 2007.
[8] G. Birkhoff, Lattice theory, American Mathematical Society Colloquium Publications, 25, 1940.
[9] T. Calvo, B. De Baets, J. Fodor, The functional equations of Frank and Alsina for uninorms and nullnorms, Fuzzy Sets and Systems, 120(3) (2001), 385-394.
[10] A. H. Clifford, Naturally totally ordered commutative semigroups, American Journal of Mathematics, 76 (1954), 631-646.
[11] A. C. Climescu, Sur l’équation fonctionelle de l’associativité, Bulletin de l’École Polytechnique de Jassy, 1, (1946), 1-16.
[12] G. P. Dimuro, B. Bedregal, Archimedean overlap functions: The ordinal sum and the cancellation, idempotency and limiting properties, Fuzzy Sets and Systems, 252 (2014), 39-54.
[13] J. C. Fodor, R. R. Yager, A. Rybalov, Structure of uninorms, International Journal of Uncertainty, Fuzziness and Knowledge-based Systems, 5 (1997), 411-127.
[14] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford, 1963.
[15] M. Grabisch, J. L. Marichal, R. Mesiar, E. Pap, Aggregation functions, Cambridge University Press, 2009.
[16] P. Hájek, Observations on the monoidal t-norm logic, Fuzzy Sets and Systems, 132 (2002), 107-112.
[17] S. Jenei, A note on the ordinal sum theorem and its consequence for the construction of triangular norms, Fuzzy Sets and Systems, 126(2) (2002), 199-205.
[18] E. P. Klement, R. Mesiar, E. Pap, Triangular norms, Kluwer Academic Publishers, Dordrecht, 2000.
[19] E. P. Klement, R. Mesiar, E. Pap, Triangular norms as ordinal sums in the sense of A.H. Clifford, Semigroup Forum, 65 (2002), 71-82.
[20] E. P. Klement, R. Mesiar, E. Pap, Archimedean components of triangular norms, Journal of the Australian Mathematical Society, 78 (2005), 239-255.
[21] C. M. Ling, Representation of associative functions, Publicationes Mathematicae Debrecen, 12 (1965), 189-212.
[22] R. Mesiar, B. De Baets, Ordinal sums of aggregation operators, In proc. of AGGREGATION ’99, Calvo T., Mesiar R. (eds.), UIB Palma de Mallorca, (1999), 133-143.
[23] R. Mesiar, A. Mesiarová, Residual implications and left-continuous t-norms which are ordinal sums of semigroups, Fuzzy Sets and Systems, 143 (2004), 47-57.
[24] R. Mesiar, C. Sempi, Ordinal sums and idempotents of copulas, Aequationes Mathematicae, 79(1-2) (2010), 39-52.
[25] A. Mesiarová-Zemánková, Multi-polar t-conorms and uninorms, Information Sciences, 301 (2015), 227-240.
[26] A. Mesiarová-Zemánková, Ordinal sum construction for uninorms and generalized uninorms, International Journal of Approximate Reasoning, 76 (2016), 1-17. [27] A. Mesiarová-Zemánková, A note on decomposition of idempotent uninorms into an ordinal sum of singleton semigroups, Fuzzy Sets and Systems, 299 (2016), 140-145.
[28] A. Mesiarová-Zemánková, Ordinal sums of representable uninorms, Fuzzy Sets and Systems, 308 (2017), 42-53.
[29] A. Mesiarová-Zemánková, Characterization of uninorms with continuous underlying t-norm and t-conorm by means of the ordinal sum construction, International Journal of Approximate Reasoning, 83 (2017), 176-192.
[30] A. Mesiarová-Zemánková, 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.
 [31] A. Mesiarová-Zemánková, Characterization of idempotent n-uninorms, Fuzzy Sets and Systems, (2020), Doi: 10.1016/j.fss.2020.12.019.
[32] A. Mesiarová-Zemánková, Characterization of n-uninorms with continuous underlying functions via z-ordinal sum construction, International Journal of Approximate Reasoning, 133 (2021), 60-79.
[33] P. S. Mostert, A. L. Shields, On the structure of semi-groups on a compact manifold with boundary, Annals of Mathematics, Series II, 65 (1957), 117-143.
[34] D. Ruiz, J. Torrens, Distributivity and conditional distributivity of a uninorm and a continuous t-conorm, IEEE Transactions on Fuzzy Systems, 14(2) (2006), 180-190.
[35] R. R. Yager, A. Rybalov, Uninorm aggregation operators, Fuzzy Sets and Systems, 80 (1996), 111-120.
[36] W. Zong, Y. Su, H. W. Liu, B. De Baets, On the structure of 2-uninorms, Information Sciences, 467 (2018), 506-527.
Iranian Journal of Fuzzy Systems
Volume 19, Issue 1
January and February 2022
Pages 83-96

Files

Share

How to cite

Statistics