On Lipschitz properties of generated aggregation functions

Authors:
Gleb Beliakov;Tomasa Calvo;Simon James
Affiliations:
School of Engineering and Information Technology, Deakin University, 221 Burwood Hwy, Burwood 3125, Australia;Departamento de Ciencias de la Computación, Universidad de Alcalá, 28871-Alcalá de Henares (Madrid), Spain;School of Engineering and Information Technology, Deakin University, 221 Burwood Hwy, Burwood 3125, Australia
Venue:
Fuzzy Sets and Systems
Year:
2010

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Abstract

This article discusses Lipschitz properties of generated aggregation functions. Such generated functions include triangular norms and conorms, quasi-arithmetic means, uninorms, nullnorms and continuous generated functions with a neutral element. The Lipschitz property guarantees stability of aggregation operations with respect to input inaccuracies, and is important for applications. We provide verifiable sufficient conditions to determine when a generated aggregation function holds the k-Lipschitz property, and calculate the Lipschitz constants of power means. We also establish sufficient conditions which guarantee that a generated aggregation function is not Lipschitz. We found the only 1-Lipschitz generated function with a neutral element e@?]0,1[.