Extremal values-based aggregation functions

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 >= 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.