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.
This paper is inspired by recent results [15, 16] which have shown that a multiplicative generator of a strict triangular norm can be reconstructed from the first partial derivatives of the triangular norm on the segment {0} x [0,1]. The strict triangular norms to which this method is applicable have been called zero-reconstructible triangular norms. This paper shows that every continuous triangular norm can be approximated (with an arbitrary precision) by a zero-reconstructible one, and thus substantiates the significance of this subclass of strict triangular norms.