The purpose of this paper is to characterize the concept of monotonicity according to a direction related to a set of n random variables in terms of its associated n-copula. We start establishing relationships in the bivariate and trivariate cases, which will help to understand the extension to the multivariate case. Several examples are provided.

In this paper, we introduce a new kind of order, Cesaro supermodular order, which includes supermodular order and stochastic order. For this new order, we show that it almost fulfils all desirable properties of a multivariate positive dependence order that have been proposed by Joe (1997). Also, we obtain some relations between it with other orders. Finally, we consider different issues related extended Archimedean copula and positive dependence.

In this study, we focus on bivariate transformations of trivariate quasi-Copulas. We characterize necessary and sufficient conditions of transformations that can be written in the form of compositions between two quasi-Copulas and a quadratic polynomial function. The conditions only depend on the coefficients of the quadratic polynomial. The set of these coefficients is convex with linear-section boundary and it lies on the seven-dimensional Euclidean space. All extreme points of this set have been characterized via CAS and can be used to construct quasi-Copulas. Construction examples are also given.