Matrix representation of discrete quasi-copulas

In this paper discrete quasi-copulas (defined on a square grid of [0,1]2) are studied and it is proved that they can be represented by means of a special class of matrices with entries in [-1,1]. Special considerations are made for the case of irreducible discrete quasi-copulas (those with range In), showing that they can be represented through alternating-sign matrices and that they generate all discrete quasi-copulas through convex sums. In the process, the number of irreducible quasi-copulas on In is given and those functions δ for which there exists a unique copula with δ as its diagonal section are characterized.