Invariant continuous aggregation functions

Abstract

Continuous aggregation functions, being invariant under any monotone bijective transformation, are discussed and characterised. As a basic tool for this characterisation, self-dual -valued capacities or their counterparts, self-dual monotone Boolean functions, are exploited. A generating role of is discussed. The relevant role of simple medians in describing monotone threshold Boolean functions is made apparent.

Keywords:

• aggregation function
• Choquet integral
• capacity
• invariant aggregation function
• self-dual capacity

Acknowledgements

Andrey Bronevich expresses his sincere thanks to the Slovak Ministry of Education, SAIA, Slovak University of Technology in Bratislava, and Prof. Radko Mesiar for making it possible to undertake this research in Slovakia. Radko Mesiar expresses his gratitude for the support of grants VEGA 1/4209/07, APVV-0012-07 and MSM VZ 6198898701.