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Strengthened ordered directionally monotone functions. Links between the different notions of monotonicity. (English) Zbl 1423.26057

Summary: In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Łukasiewicz implication.

MSC:

26E50 Fuzzy real analysis
26A48 Monotonic functions, generalizations
28E10 Fuzzy measure theory
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References:

[1] Beliakov, G.; Bustince, H.; Calvo, T., A Practical Guide to Averaging Functions, Stud. Fuzziness Soft Comput. (2016), Springer International Publishing
[2] Beliakov, G.; Calvo, T.; Wilkin, T., Three types of monotonicity of averaging functions, Knowl.-Based Syst., 72, 114-122 (2014)
[3] Beliakov, G.; Špirková, J., Weak monotonicity of Lehmer and Gini means, Fuzzy Sets Syst., 299, 26-40 (2016) · Zbl 1416.68053
[4] Bustince, H.; Barrenechea, E.; Pagola, M., Restricted equivalence functions, Fuzzy Sets Syst., 157, 17, 2333-2346 (2006) · Zbl 1110.68158
[5] Bustince, H.; Barrenechea, E.; Sesma-Sara, M.; Lafuente, J.; Dimuro, G. P.; Mesiar, R.; Kolesárová, A., Ordered directionally monotone functions. Justification and application, IEEE Trans. Fuzzy Syst. (2017)
[6] Bustince, H.; Fernandez, J.; Kolesárová, A.; Mesiar, R., Directional monotonicity of fusion functions, Eur. J. Oper. Res., 244, 1, 300-308 (2015) · Zbl 1346.26004
[7] De Miguel, L.; Sesma-Sara, M.; Elkano, M.; Asiain, M.; Bustince, H., An algorithm for group decision making using n-dimensional fuzzy sets, admissible orders and owa operators, Inf. Fusion, 37, 126-131 (2017)
[8] García-Lapresta, J.; Martínez-Panero, M., Positional voting rules generated by aggregation functions and the role of duplication, Int. J. Intell. Syst., 32, 9, 926-946 (2017)
[9] Grabisch, M.; Marichal, J.; Mesiar, R.; Pap, E., Aggregation Functions (2009), Cambridge University Press · Zbl 1196.00002
[10] Lucca, G.; Sanz, J.; Dimuro, G.; Bedregal, B.; Asiain, M. J.; Elkano, M.; Bustince, H., CC-integrals: Choquet-like copula-based aggregation functions and its application in fuzzy rule-based classification systems, Knowl.-Based Syst., 119, 32-43 (2017)
[11] Lucca, G.; Sanz, J. A.; Dimuro, G. P.; Bedregal, B.; Bustince, H.; Mesiar, R., CF-integrals: a new family of pre-aggregation functions with application to fuzzy rule-based classification systems, Inf. Sci., 435, 94-110 (2017)
[12] Lucca, G.; Sanz, J. A.; Dimuro, G. P.; Bedregal, B.; Mesiar, R.; Kolesárová, A.; Bustince, H., Preaggregation functions: construction and an application, IEEE Trans. Fuzzy Syst., 24, 2, 260-272 (2016)
[13] Mesiar, R.; Kolesárová, A.; Stupňanová, A., Quo vadis aggregation?, Int. J. Gen. Syst., 47, 2, 97-117 (2018)
[14] Paternain, D.; Fernandez, J.; Bustince, H.; Mesiar, R.; Beliakov, G., Construction of image reduction operators using averaging aggregation functions, Fuzzy Sets Syst., 261, 87-111 (2015) · Zbl 1360.68881
[15] Sesma-Sara, M.; Bustince, H.; Barrenechea, E.; Lafuente, J.; Kolesárová, A.; Mesiar, R., Edge detection based on ordered directionally monotone functions, (Advances in Fuzzy Logic and Technology 2017 (2018), Springer International Publishing), 301-307
[16] Wilkin, T.; Beliakov, G., Weakly monotonic averaging functions, Int. J. Intell. Syst., 30, 2, 144-169 (2015)
[17] Yager, R. R., On ordered weighted averaging aggregation operators in multicriteria decisionmaking, IEEE Trans. Syst. Man Cybern., 18, 1, 183-190 (1988) · Zbl 0637.90057
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