Bivariate p-boxes and maxitive functions

We give necessary and sufficient conditions for a maxitive function to be the upper probability of a bivariate p-box, in terms of its associated possibility distribution and its focal sets. This allows us to derive conditions in terms of the lower and upper distribution functions of the bivariate p-box. In particular, we prove that only bivariate p-boxes with a non-informative lower or upper distribution function may induce a maxitive function. In addition, we also investigate the extension of Sklar's theorem to this context.