On a class of transformations of copulas and quasi-copulas

The theory of copulas is by now a very well established one. Recently, larger classes of functions C:[0,1]n→[0,1], that are increasing in each variable and satisfy some conditions at the boundary (like quasi-copulas), have been the object of fruitful research. Several authors have considered the action of a class of transformations on some aggregation operators, as t-norms, copulas, quasi-copulas and so on. These simple transformations do not preserve in general all properties of copulas (or quasi-copulas): in particular, the fact that only some properties are preserved by these transformations, suggested the introduction of semi-copulas. The purpose of the present contribution is to give a fairly complete picture concerning such action on copulas and quasi-copulas; in particular, we prove results concerning inclusions and strict inclusions among these classes of operators, and those of their transforms.