M¨ obius representation of the bipolar decomposition integrals and its applications

Abbas, J.; Mesiar, R.; Halas, R.

doi:10.22111/ijfs.2025.9334

Document Type : Research Paper

Authors

¹ Department of Applied Sciences, University of Technology, Al Sina´ a Street, P. O. Box (19006), Baghdad, Iraq

² Faculty of Civil Engineering, Slovak University of Technology, Radlinsk´ eho 11, 810 05 Bratislava, Slovakia

³ Department of Algebra and Geometry, Palack´y University of Olomouc, 17. listopadu 12, 779 00 Olomouc, Czech Republic

https://doi.org/10.22111/ijfs.2025.9334

Abstract

The bipolar decomposition integral, recently introduced in [5], is a general framework for handling integrals related
to aggregation on unipolar and bipolar scales. The M¨obius representation is related to the notion of k-additivity
of a monotone set function and allows to derive simple expressions of some nonlinear integrals. In this paper, we
propose M¨obius representation for the bipolar decomposition integral, which includes the M¨obius representation of
each of the bipolar Choquet integral, bipolar Shilkret integral, and bipolar Pan integral. Then, we introduce the
expressions for computing bipolar decomposition integrals concerning a 2-additive bi-capacity. Lastly, a practical
numerical example is provided to illustrate the applicability of the proposed results in dealing with aggregation on
bipolar scales, and simplicity of calculating the 2-additive bipolar decomposition integrals.

Keywords

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M¨ obius representation of the bipolar decomposition integrals and its applications

References

References

Volume 22, Issue 4July and August 2025Pages 1-15

Volume 22, Issue 4
July and August 2025
Pages 1-15