Extremal values-based aggregation functions

Abstract

We introduce and study aggregation functions based on extremal values, namely extended $(l,u)$-aggregation functions whose outputs only depend on a fixed number l of extremal lower input values and a fixed number u of extremal upper input values, independently of the arity of the input n-tuples ($n\ge l+u$). We discuss several general properties of $(l,u)$-aggregation functions and we study special $(l,u)$-aggregation functions with neutral element, including t-conorms, t-norms and uninorms. We also study $(l,u)$-aggregation functions defined by means of integrals with respect to discrete fuzzy measures, as well as $(l,u)$-ordered weighted quasi-arithmetic means based on appropriate weighting vectors. We also stress some generalizations based on recently introduced new types of monotonicity. Some possible applications are sketched, too.