The unwalked path between quasi-copulas and copulas: Stepping stones in higher dimensions

We show that as the dimensionality increases, more and more interesting classes of operations can be identified between the class of n-quasi-copulas and the class of n-copulas. One such class is the class of supermodular n-quasi-copulas. We observe that some properties of 2-copulas that cannot be generalized to higher-dimensional copulas, hold true for supermodular n-quasi-copulas. Additionally, we show that trying to generalize a volume-based characterization of bivariate copulas to higher dimensions, results in a restrictive class of n-quasi-copulas.