k-Additivity and C-decomposability of bi-capacities and its integral

k-Additivity is a convenient way to have less complex (bi-)capacities. This paper gives a new characterization of k-additivity, introduced by Grabisch and Labreuche, of bi-capacities and contrasts between the existing characterization and the new one, that differs from the one of Saminger and Mesiar. Besides, in the same way for capacities, a concept of C-decomposability, distinct from the proposal of Saminger and Mesiar, but closely linked to k-additivity, is introduced for bi-capacities. Moreover, the concept of C-decomposability applies to the Choquet integral with respect to bi-capacities.