Characterizations of Archimedean $n$-copulas

summary:We present three characterizations of $n$-dimensional Archimedean copulas: algebraic, differential and diagonal. The first is due to Jouini and Clemen. We formulate it in a more general form, in terms of an $n$-variable operation derived from a binary operation. The second characterization is in terms of first order partial derivatives of the copula. The last characterization uses diagonal generators, which are ``regular'' diagonal sections of copulas, enabling one to recover the copulas by means of an asymptotic representation