Extremal Lipschitz continuous aggregation functions with a given diagonal section

In this paper we study the smallest and the greatest M-Lipschitz continuous n-ary aggregation functions with a given diagonal section. We show that several properties that were studied for the smallest and the greatest 1-Lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while considering different Lipschitz constants. Just as in the binary case, we show that the smallest n-quasi-copula with a given diagonal section coincides with the smallest 1-Lipschitz n-ary aggregation function with that diagonal section. Additionally, we show that the smallest n-quasi-copula with a given diagonal section, called the Bertino n-quasi-copula, is supermodular for any n⩾2.